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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87% (01:54) correct
