Divisibility

jsasipriya wrote:Is N^2-N divisible by 12 ?

1. N is divisible by 11
2. N is divisible by 19

(N² - N) = N(N - 1)

Statement 1: N is divisible by 11.
Implies (N² - N) is divisible by 11. But (N² - N) may or may not be divisible by 12 depending upon whether the product of N and (N - 1) is divisible by 12 or not. For example consider the following cases,

• 1. N = 11 => (N² - N) = N(N - 1) = 11*10 => NO
  2. N = 33 => (N² - N) = N(N - 1) = 33*32 = (3*11)*(4*8) => YES

Not sufficient

Statement 2: N is divisible by 19.
Implies (N² - N) is divisible by 19. But (N² - N) may or may not be divisible by 12 depending upon whether the product of N and (N - 1) is divisible by 12 or not. For example consider the following cases,

• 1. N = 19 => (N² - N) = N(N - 1) = 19*18 => NO
  2. N = 57 => (N² - N) = N(N - 1) = 57*56 = (3*19)*(4*14) => YES

Not sufficient

1 & 2 Together: N is divisible by 11 and 19. But (N² - N) may or may not be divisible by 12 depending upon whether the product of N and (N - 1) is divisible by 12 or not.

Not sufficient.

The correct answer is E.

Note: There is no need to find examples to prove the insufficiency of the statements. It's quiet obvious that N being divisible by 11 and/or 19 does not ensure that N(N - 1) is divisible by 12. I have just provided them in case there is any confusion.

jsasipriya wrote:Thank you very much for your explanation.

Could you please explain your statement "It's quiet obvious that N being divisible by 11 and/or 19 does not ensure that N(N - 1) is divisible by 12." little more?

Without picking numbers, how can we say whether N(N-1) is divisible by 12 or not?

Because if N is divisible by 11, there is no guarantee that either N or (N - 1) or their product N(N - 1) will be divisible by 12. The information N is divisible by 11 only allow us to conclude that N(N - 1) is also divisible by 11. But for a number to be divisible by 12, it must be divisible by 3 and 4. But 11 being a prime, divisibility by 11 ensures divisibility by 11 only.

Same goes for 19 also.

I think it'll be easier for me to explain if you let me know what your understanding is.