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What is the units digit for the number 123^(456) ?

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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.

Weekly Problem 10 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1
You could write this problem out as 1x 3 x 4x 6 x 7 x 8 x 9 x (2 x 5) x 10. The product of the first seven numbers is not a multiple of ten, but the last part gives us 10 x 10 so the overall product must be a multiple of 100, but not a multiple of 1000. So the answer is two zeros.

This problem is taken from the UKMT Mathematical Challenges.

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