Factoring factorials
Find the highest power of 11 that will divide into 1000! exactly.
Age
11 to 14
Challenge level
Problem
Find the highest power of 11 that will divide 1000! exactly.
[Factorial 1000, written 1000!, is the product of the first 1000 whole numbers 1 x 2 x 3 x ... x 1000]
Student Solutions
My name is Talei and I am a pupil from Poltair Community School and Sports College in St Austell, in Cornwall.
The highest power of 11 which will divide exactly into 1000! is 11 98
I worked this out by:-
- deciding that there are 90 multiples of 11 from 11 to 990 multiplied within 1000!
- in a fraction with all the factors of 1000! as the numerator and with a denominator of as many elevens as possible to cancel out the multiples of 11 in the numerator, you would cancel out 90 elevens from every multiple of 11, e.g. 22/11= 2, and a further eight elevens from each multiple which could be divided by eleven twice, e.g. 11 x 11, 22 x 11, 33 x 11 up to 88 x 11
- and turning each eleven into a power, gives my above conclusion.
- This would definitely divide exactly into 1000!
Well done Talei! Congratulations also to Bethany, Emma and Monica of Hethersett High School and Soh Yong Sheng, of Raffles Institution, Singapore.
Teachers' Resources
Possible extension.
This problem is a special case of Factorial Fun.
Possible support
See also Fac Finding and Powerful Factorialwhich are also special cases.
This problem is a special case of Factorial Fun.
Possible support
See also Fac Finding and Powerful Factorialwhich are also special cases.