Find the Maximum Power of 15 in 75!

June 23, 2019

Find the Maximum Power of 15 in 75!

We should commit to the memory that the above method is applicable only to prime numbers.  But 15 is a composite number.  (15 = 3 x 5). So we find the maximum power of 3 and 5 in the above expression.
Important note: The maximum power of 5 in the above expression is less than the maximum power of 3 as 5 is bigger number.

From the above diagram, there are 24 fives and 48 threes available.  The number of 15’s can be formed = 24.
( Shortcut: To remember easily, think like, there are 24 women, and 48 men available. How many legal marriages possible? answer is 24! )
Theoretically, $100!=3^{48}\times 5^{24}\times ...$ = $3^{24}\times \left(3^{24}\times 5^{24}\right)\times ...$ = $3^{24}\times \left({15}^{24}\right)\times ...$
So maximum power of 15 is 24