power of a number in a factorial

Question

what is the largest power of 24 in 150! ?

HINT : answer is 48

I need to know the method for solving such questions when the highest power of the number to be found is non-prime ..

Answer 1

You have to compute the highest power of each prime factor of the number you want (for 24, these are 2 and 3) and then for each of these, compute how many times that power of the primes divide the factorial (for 24 it is $2^3$ and $3^1$), and choose the smaller.

For your case, 2 goes 75+37+18+9+4+2+1=146, so $2^3$ goes $[146/3] = 48$. 3 goes 50+16+5+1 = 73. The smaller of these is 48.

Answer 2

I assume from what you wrote that you know how to find the highest power of a prime $p$ that divides $n!$.

Compute $a$, the largest number such that $2^a$ divides $150!$.

Calculate $b$, the largest number such that $3^b$ divides $150!$.

Which is smaller, $a/3$ or $b$? We are comparing these because we need three $2$'s for every $3$.

That tells you what the limiting resource is, $2$'s or $3$'s.