Question

Let a,b be nonnegative integers. Compute the number of positive divisors of 2^a x 3^b. Prove your answer

Answer 1

\[2^{a}\times3^{b}\]

Answer 2

Let:\[n=2^a\cdot3^b\] an integer will divide n when it is of the form:\[2^x\cdot3^y,x\leq a,y\leq b\]so there are a+1 choices for the exponent x (the numbers 0 to a), and b+1 choices for the exponent of y. Therefore there are (a+1)(b+1) divisors of n.

Answer 3

thank you!