Counting Factors

Is there an efficient way to work out how many factors a large number has?

Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Factoring Factorials

Stage: 3 Challenge Level:

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.