teal wrote:

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

2ab is divisible by 5

a - b is divisible by 5

I have a question regarding this question. I want to verify the property of multiples that is related to divisibility. So

a) Mutliple of N + Mutliple of N = Multiple of N

b) Mutliple of N + Non -Mutliple of N = Non-Multiple of N

c) Non-Mutliple of N +Non- Mutliple of N = Can be both (multiple or non-multiple)

Does this property work for all integers, are there any exceptions?

There are no exceptions.

If integers a and b are both multiples of some integer k>1 (divisible by k), then their sum and difference will also be a multiple of k (divisible by k):Example:

a=6 and

b=9, both divisible by 3 --->

a+b=15 and

a-b=-3, again both divisible by 3.

If out of integers a and b one is a multiple of some integer k>1 and another is not, then their sum and difference will NOT be a multiple of k (divisible by k):Example:

a=6, divisible by 3 and

b=5, not divisible by 3 --->

a+b=11 and

a-b=1, neither is divisible by 3.

If integers a and b both are NOT multiples of some integer k>1 (divisible by k), then their sum and difference may or may not be a multiple of k (divisible by k):Example:

a=5 and

b=4, neither is divisible by 3 --->

a+b=9, is divisible by 3 and

a-b=1, is not divisible by 3;

OR:

a=6 and

b=3, neither is divisible by 5 --->

a+b=9 and

a-b=3, neither is divisible by 5;

OR:

a=2 and

b=2, neither is divisible by 4 --->

a+b=4 and

a-b=0, both are divisible by 4.

As for the question:

If a and b are positive integers, is a^2+b^2 divisible by 5 ?(1)

2ab is divisible by 5 --> if

a=b=5 then the answer is YES but if

a=5 and

b=1 then the answer is NO. Not sufficient.

(2)

a-b is divisible by 5 --> if

a=b=5 then the answer is YES but if

a=b=1 then the answer is NO. Not sufficient.

(1)+(2) From (2)

a-b is divisible by 5 so

(a-b)^2=(a^2+b^2)-2ab is also divisible by 5. Next, since from (1)

2ab is divisible by 5 then

a^2+b^2 must also be divisible by 5 in order their sum to be divisible by 5. Sufficient.

Answer: C.

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