If a and b are positive integers, is a^2 + b^2 divisible by 5?

Yes no type Qs, a
Given info- a and b are positive integer then a2+b2/5 =integer??

St-1 2ab is divisible by 5, then a is multiple of 5 or b is multiple of 5 or both a & b is multiple of 5.
If a& b both multiple of 5 then sufficient if a only or b only is multiple then insufficient.

St-2 a-b is multiple of 5, a and b both multiple of 5 then sufficient also a and b can not be multiple of 5 eg. 8-3 is divisible by 5. insufficient.

Combine--If either a or b is multiple of 5 then other one has to be multiple of 5 to satisfy a-b is multiple of 5. then sufficient.
if a is 10 and b is 2 then a-b can not be divisible by 5. for both statement true and a and b has to be multiple of 5.

Answer is C

Solution:
Pre Analysis:

• \(a\) and \(b\) are positive integers
• We are asked if \(a^2 + b^2\) divisible by 5 or not

Statement 1: \(2ab\) is divisible by 5

• According to this statement, \(2ab=5k\) where k is a positive integer
• However, this doesn't tell us if \(a^2+b^2\) is divisible by 5 or not
• Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: \(a - b\) is divisible by 5

• According to this statement, \(a-b=5q\) where q is a positive integer
• Squaring both sides we have \((a-b)^2=(5q)^2\)
  \(⇒a^2+b^2-2ab=25q^2\)
  \(⇒a^2+b^2=25q^2+2ab\)
• However, this doesn't tell us if \(a^2+b^2\) is divisible by 5 or not without knowing if \(2ab\) is divisible by 5 or not
• Thus, statement 2 alone is not sufficient

Combining:

• From statement 1, \(2ab=5k\)
• From statement 2, \(a^2+b^2=25q^2+2ab\)
• So, \(a^2+b^2=25q^2+2ab=25q^2+5k=5(5q^2+k)\)
• Which means \(a^2+b^2\) is divisible by 5

Hence the right answer is Option C

If \(a\) and \(b\) are positive integers, is \(a^2 + b^2\) divisible by 5?

(1) \(2ab\) is divisible by 5.

This implies that either \(a\), \(b\), or both are divisible by 5. If \(a = b = 5\), the answer is YES. However, \(a = 5\) and \(b = 1\), the answer is NO. Not sufficient.

(2) \(a-b\) is divisible by 5.

This implies that either both \(a\) and \(b\) are divisible by 5 or neither are. If \(a = b = 5\), the answer is YES. However, \(a = b = 1\), the answer is NO. Not sufficient.

(1)+(2) From statement (2), \(a - b\) is divisible by 5, so \((a-b)^2=(a^2+b^2)-2ab\) is also divisible by 5. Since from statement (1) \(2ab\) is divisible by 5, then \(a^2 + b^2\) must also be divisible by 5 for their difference to be divisible by 5. Sufficient.

Alternatively, from statement (2), if neither \(a\) nor \(b\) is divisible by 5, it would contradict statement (1). Thus, the only remaining scenario from statement (2) is that both \(a\) and \(b\) are divisible by 5. Consequently, \(a^2 + b^2\) is also divisible by 5. Sufficient.


Answer: C