rheam25 wrote:
If M and N are integers such that M = \(N^2\) – 1, is M divisible by 12?
(1) N-1 is the square of an even number
(2) N+1 is a two digit number and the sum of its digits is a multiple of 3
M = N^2 - 1
M = (N-1) (N+1)
We need to check whether (N-1)(N+1) contains 2^2 and 3.Statement-1:N-1 is the square of an even number.
Therefore, (N-1) = (2k)^2 where k is an integer.
So, (N-1) contains 2^2.
Does (N-1) or (N+1) contains 3? Let's check.
If k=3, then (N-1) = 36 and (N+1) = 14. Here, M is divisible by 12.
If k=4, then (N-1) = 64 and (N+1) = 66. Here, M is not divisible by 12.
There may or may not be 3 in (N-1)(N+1).
So, statement 1 is not sufficient alone.
Statement-2:N+1 is a two digit number and the sum of its digits is a multiple of 3
(N+1) is divisible by 3.
Does (N-1) or (N+1) contains 2^2? Let's check.
If (N+1)=12, then (N-1)=10. Here, M= (N-1)(N+1)= 12*10. M is divisible by 12.
If (N+1)=15, then (N-1)=13. Here, M= (N-1)(N+1)= 15*13. M is not divisible by 12.
There may or may not be 2^2 in (N-1)(N+1).
So, statement 2 is not sufficient alone.
Combining both statements:N-1 is the square of an even number.
N+1 is a two digit number and the sum of its digits is a multiple of 3
(N-1) contains 2^2 and (N+1) contains 3.
So, M = (N-1)(N+1) contains 2^2 and 3.
Hence, M is divisible by 12.
Therefore, both statements together are sufficient. Answer is C.
The highlighted part above is divisible by 12.