Question

find the number of positive integers less than 101 that can not be written as the difference of two squares of integers.

Answer 1

Answer:

25

Step-by-step explanation:

The two things here to be noted are:

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

Answer 2

Answer:

25

Step-by-step explanation:

The two things here to be noted are:

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^2m=(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)^2m=(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

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