goelmohit2002 wrote:goelmohit2002 wrote:Will this above method work for all prime no.s ?
For e.g. if we are required to find out max value of k in 144! for
a) 2^k
b) 3^k
c) 31^k
Will this method always work for all prime numbers ?
Thanks
Mohit
Experts please share your thoughts about the above....
Dana gave an excellent explanation above of why you should *only* apply this method after breaking your number into primes, but yes, it will work for any prime number. So if we want to prime factorize say 20!, we can do that quickly enough:
20/2 = 10; 20/4 = 5; 20/8 ~ 2 (round down); 20/16 ~ 1 (round down)
so 2^18 is the largest power of 2 which divides 20!
Continuing for 3, and rounding down with each division:
20/3 ~ 6; 20/9 ~ 2
so 3^8 is the largest power of 3 which will divide 20!
We also have 20/5 = 4, and 20/7 ~ 2, and the remaining primes only divide 20! once, giving us:
20! = 2^(18) * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19
That's not the type of thing you'd normally need to do on the GMAT, but if you can prime factorize a number like 20! quickly, then questions on this topic should be fairly easy to answer.
Finally, since this thread seems to have attracted some interest, I'll offer three questions which are variations on the question in the original post.
You'd only see questions like the below at the 750-800 level of the test, and while I've never seen real GMAT questions similar to questions 2 or 3 below, they both can be solved very quickly once you see what to do.
1. What is the largest positive integer k for which 27^k is a divisor of 45! ?
2. If z is equal to the product of the first 12 positive
even integers, what is the largest integer k for which 2^k is a factor of z?
3. What is the largest positive integer k for which 2^k is a divisor of 9! - 8! ?
Enjoy!
