BdMO Online Forum

A new series is to be created after erasing some numbers from the series 1, 2, 3,
4…………. 400, in such a way so that the sum of any two numbers from the series
are not divisible by 7. What is the maximum number of terms to be found in new
series?

source:2014 dhaka ancholik, junior

bhai jan the ans will be probably 2.......
from iftehazyeasir

We can divide all the numbers in 7 parts as (mod 7),7n,7n+1,7n+2,7n+3,7n+4,7n+5,7n+6.
now see, 7n+1+7n+6=14n+7,7n+2+7n+5=14n+7 and 7n+3+7n+4=14n+7.so if we take those number which are 1(mod 7) we cannot take those which are 6(mod 7) and so on.
there are floor of (400-1)/7 or 57 numbers from 1 to 400 which are 1 (mod 7)
floor of (400-2)/7 or 56 numbers of 2(mod 7)
floor of (400-3)/7 or 56 numbers of 3 (mod 7). But see if we take a number which is 0 (mod 7) there is no number which can make a sum divisible by 7.so total =57+56+56+1=170.