We will prove this by induction.
But first, notice that the set of odd digits <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:5.339em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char66.png" style="height:15px; width:6px; vertical-align:-4px; margin-right:0.068em;"><span class="icmr10">1</span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">3</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">5</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">7</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">9</span></span><img src="
jsMath/fonts/cmsy10/alpha/100/char67.png" style="height:15px; width:6px; vertical-align:-4px; margin-right:0.068em;"> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.104em;vertical-align:-0.3em"></span></span></nobr></span> is a complete set of residue modulo <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.526em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span>
Base case : When <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:2.105em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">n</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">1</span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span>, We have <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.526em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">1</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span> digit number <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.526em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span> which is divisable by <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.902em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.944em;vertical-align:-0.05em"></span></span></nobr></span>.
Induction hypothesis : Let's assume we have <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:2.105em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">n</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">k</span><span class="spacer" style="margin-left:0.031em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span> digit number <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:8.498em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmsy10">−</span><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.277em;vertical-align:-0.336em"></span></span></nobr></span> where <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.752em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">i</span></span><span class="spacer" style="margin-left:0.05em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.076em;vertical-align:-0.318em"></span></span></nobr></span>s are odd digits.
Inductive step : We claim that there exists an odd digit <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:1.654em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.141em;vertical-align:-0.336em"></span></span></nobr></span> such that <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:4.888em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.141em;vertical-align:-0.336em"></span></span></nobr></span> is divisable by <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:1.654em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.990em;vertical-align:-0.05em"></span></span></nobr></span>.
Proof : Notice that, if <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:13.537em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">2</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">x</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char11.png" style=" width:11px; margin-right:-0.013em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">2</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:1em;"><span class="icmr10">(</span></span><span class="icmr10">mod</span><span style="position: relative; margin-left:0.333em;"><span class="icmr10">5</span></span><span class="icmr10">)</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.277em;vertical-align:-0.336em"></span></span></nobr></span>,
Then, <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:6.543em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char11.png" style=" width:11px; margin-right:-0.013em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:1em;"><span class="icmr10">(</span></span><span class="icmr10">mod</span><span style="position: relative; margin-left:0.333em;"><span class="icmr10">5</span></span><span class="icmr10">)</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.054em;vertical-align:-0.25em"></span></span></nobr></span>
So, from a complete set <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.601em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">S</span><span class="spacer" style="margin-left:0.057em"></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span> of residue modulo <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.526em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span>, there must exist <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:2.256em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">S</span><span class="spacer" style="margin-left:0.057em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.893em;vertical-align:-0.089em"></span></span></nobr></span> such that <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:10.077em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">2</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">x</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char11.png" style=" width:11px; margin-right:-0.013em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span><span style="position: relative; margin-left:1em;"><span class="icmr10">(</span></span><span class="icmr10">mod</span><span style="position: relative; margin-left:0.333em;"><span class="icmr10">5</span></span><span class="icmr10">)</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.277em;vertical-align:-0.336em"></span></span></nobr></span>
So, there exists <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:8.197em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char66.png" style="height:15px; width:6px; vertical-align:-4px; margin-right:0.068em;"></span><span class="icmr10">1</span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">3</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">5</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">7</span></span><img src="
jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmr10">9</span></span><img src="
jsMath/fonts/cmsy10/alpha/100/char67.png" style="height:15px; width:6px; vertical-align:-4px; margin-right:0.068em;"> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.141em;vertical-align:-0.336em"></span></span></nobr></span> such that <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:5.038em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">2</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.277em;vertical-align:-0.336em"></span></span></nobr></span> is divisable by <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:0.526em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.854em;vertical-align:-0.05em"></span></span></nobr></span>
Now, <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:33.091em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">1</span></span><span class="icmr10">0</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.859em;"><img src="
jsMath/fonts/cmex10/alpha/100/char00.png" style="height:18px; width:6px; vertical-align:-17px; margin-right:0.026em;"></span><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmsy10">−</span><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="
jsMath/fonts/cmmi10/alpha/100/char3A.png" style=" width:3px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmr10">1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; top:-0.859em;"><img src="
jsMath/fonts/cmex10/alpha/100/char01.png" style="height:18px; width:5px; vertical-align:-17px; margin-right:0.098em;"></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">1</span></span><span class="icmr10">0</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.166em;"><span style="position: relative; top:-0.859em;"><img src="
jsMath/fonts/cmex10/alpha/100/char00.png" style="height:18px; width:6px; vertical-align:-17px; margin-right:0.026em;"></span><span class="icmr10">2</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="
jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">b</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">a</span><span style="position: relative; top:0.286em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; top:-0.859em;"><img src="
jsMath/fonts/cmex10/alpha/100/char01.png" style="height:18px; width:5px; vertical-align:-17px; margin-right:0.098em;"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.3em;vertical-align:-0.350em"></span></span></nobr></span>, which is divisable by <span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:1.654em"><span style="position:relative;"><span style="position:absolute; top:-0.097em; left:0em;"><span class="icmr10">5</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmmi10">k</span></span><span class="spacer" style="margin-left:0.022em"></span><span class="size2"><span class="icmr10">+1</span></span><span class="spacer" style="margin-left:0.05em"></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:0.990em;vertical-align:-0.05em"></span></span></nobr></span>
(Proved)