Question

Find the number of positive integers less than 101 that can not be written as we
of two squares of integers​

Answer 1

Answer:

25

Step-by-step explanation:

It is worth noticing that:

1) Every odd number can be written as difference of square of two integers

Let m be an odd number

then m can be written as

$m=2k+1$

or, $m=(k+1)^2-k^2$

2) Every even number that is a multiple of 4 can be written as difference of square of two integers

Let m be an even number multiple of 4 then

$m=4k$

or, $m=(k+1)^2-(k-1)^2$

Total number of positive odd integers less than 101 (i.e. 101 is not included) = 50

Total number of positive integers divisible by 4 = 25

Therefore, total positive integers less than 101 that can be written as difference of squares of two positive integers = 50 + 25 = 75

Thus, total positive integers less than 101 that cannot be written as difference of squares of two positive integers = 100 - 75 = 25

Hope this is helpful.