In the prime factorization of 109!, what is the exponent of 7? (Reminder: The number n! is the product of the integers from 1 to n. For example, 5! = 5 * 4 * 3 * 2* 1 = 120.)
+2401 Hello Guest!
We're kinda looking for the number of 7s in the numbers 1-109.
The amount of numbers divisable by 7 from 1-109 is 15 (7, 14, 21, 28, ..., 105).
However, some numbers are divisable 49 which is made out of 2 7s, 7^2.
The amount of numbers divisable by 49 from 1-109 is 2 (49, 98).
So there are 15 numbers divisable by 7, 2 of which are divisable by 49.
There are 13 numbers divisable by 7 and not 49, and 2 number divisable by 49.
Thus, our answer is 13 + 2*2= 17.
A cool way to do this is to just keep dividing.
Btw, floor is just the highest integer lower than the answer. For example floor(3.5) = 3.
floor(109/7) = 15
floor(15/7) = 2
floor(2/7) = 0
We stop dividing when our final answer is less than 1.
Then, we just add everything, 15 + 2 = 17.
This works because dividing by 7 two times is just like dividing by 49.
I hope this helped. :)))))
=^._.^=
+2401 Hello Guest!
We're kinda looking for the number of 7s in the numbers 1-109.
The amount of numbers divisable by 7 from 1-109 is 15 (7, 14, 21, 28, ..., 105).
However, some numbers are divisable 49 which is made out of 2 7s, 7^2.
The amount of numbers divisable by 49 from 1-109 is 2 (49, 98).
So there are 15 numbers divisable by 7, 2 of which are divisable by 49.
There are 13 numbers divisable by 7 and not 49, and 2 number divisable by 49.
Thus, our answer is 13 + 2*2= 17.
A cool way to do this is to just keep dividing.
Btw, floor is just the highest integer lower than the answer. For example floor(3.5) = 3.
floor(109/7) = 15
floor(15/7) = 2
floor(2/7) = 0
We stop dividing when our final answer is less than 1.
Then, we just add everything, 15 + 2 = 17.
This works because dividing by 7 two times is just like dividing by 49.
I hope this helped. :)))))
=^._.^=
