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# Factoring Factorials

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Age 11 to 14

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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.