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Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

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Counting Factors

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

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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.

Dozens

Age 7 to 14 Challenge Level:

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
 
In the interactivity below, the computer generates two random digits.
Your task is to find the largest possible three-digit number which uses the computer's digits, and one of your own, to make a multiple of 2, 3, 4 or 6.

Can you decribe a strategy that ensures your first 'guess' is always correct?
 

Enter the biggest three-digit multiple of you can think of that uses the digits:
and


Something else to think about:
 
What is the largest possible five-digit number
divisible by $12$ that you can make from the digits
$1$, $3$, $4$, $5$ and one more digit? 
 
Click here for a poster of this problem.