If m and n are both two digit numbers and m-n = 11x, is x an integer?The question basically asks whether m-n is a multiple of 11.
(1) The tens digit and the units digit of m are same --> m could be: 11, 22, 33, ..., 99 --> m is a multiple of 11. Not sufficiient since no info about n.
(2) m+n is a multiple of 11 --> if m=n=11 then the m-n is a multiple of 11 but if m=12 and n=10 then m-n is NOT a multiple of 11. Not sufficient.
(1)+(2) From (1) we have that m={multiple of 11} and from (2) we have that m+n={multiple of 11} --> {multiple of 11}+n={multiple of 11} --> n={multiple of 11} --> m-n={multiple of 11}-{multiple of 11}={multiple of 11}. Sufficient.
Answer: C.
Below might help to understand this concept better.
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.
Hope it's clear.
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