## Sylhet - 2014

For students of class 9-10 (age 14-16)

### Sylhet - 2014

A book has some page missing consecutively. The sum of the page number of
missing pages is \${976}\$. How many pages are missing there?
Mahfuz Sobhan

Posts: 25
Joined: Sat Feb 07, 2015 5:40 pm

### Re: Sylhet - 2014

Let's find out the sum of some consecutive numbers greater than 976 with the lowest difference. That is:
\$1+2+3+...+44=(41*44)/2=990\$
990-976=14 ; the sum of the missing page number.

Look, \$14=2+12=2+3+9=2+3+4+5\$ ; that we want.
So, there were 4 pages missing.
[Maybe it is the easiest way ]

Tasnood

Posts: 11
Joined: Tue Jan 06, 2015 1:46 pm

### Re: Sylhet - 2014

You're actually misinterpreting the question.

The problem basically gives us that \$(x+1) + (x+2) + ... + (x+y) =976 \$. With \$x+1\$ being the first missing page and \$x+y\$ the last missing page. So we need to find \$y\$.

\$(x+1) + (x+2) + ... + (x+y) = xy + \dfrac {y(y+1)}{2}=976\$
So, \$y(x+ \dfrac {y+1}{2})=976\$.
Now, \$\dfrac {y+1}{2}\$ is an integer. So \$y\$ must be odd. Let's look at the divisors of \$976\$. They are \$1,2,4,8,16,61,122,244,488,976\$. The question mentioned several pages so \$y \neq 1\$ and \$y\$ is odd and the only odd number except \$1\$ in the set is \$61\$. So we get, \$y=61\$

And there are \$61\$ pages.
Frankly, my dear, I don't give a damn.

Posts: 147
Joined: Mon Mar 28, 2016 6:21 pm