If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +

\(x(x + 1)\) is a multiple of 12 and 9 MEANS multiple of LCM(12,9) or 36
x and x+1 are consecutive numbers, so at the maximum, only one one will be a multiple of 3.
So we are looking at x or x+1 being divisible by 9, and then check if the even number in x and x+1 is divisible by 4
(I) Odd multiple of 9
Whenever there is an odd multiple of 9, one of the even digits on two sides will be divisible by 2 and the other by 4. Hence all odd multiple of 9 will qualify..
9, 27, 45, 63, 81, 99- 6 ways
(II) Even multiple of 9
(a) Whenever the even multiple is just by 2..NO, so eliminate 9*2, 9*6
(b) Whenever the even multiple is by 4..the number itself is completely divisible by 36, so we get two values where x or x+1 is divisible by 9.. for example 35*36 or 36*37.
so 9*4, 9*8 will give TWO each - 4 ways

Total -- 6+4=10..
Possible values of x = 1 to 100

Probability -- \(\frac{10}{100}=\frac{1}{10}\)

A