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N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

396

Stage: 4 Challenge Level: Challenge Level:1

The article on divisibility tests is very helpful when solving this problem.

  • Let the number be 3 a 1 b 4 c 0 d 92 where a, b, c and d are 5, 6, 7 and 8 in any order.
  • 396 is the product 4 x 9 x 11 so if the number is divisible by all of 4, 9 and 11 then it must be divisible by 396.
  • Since the last 2 digits are 92, the number must be divisible by 4 (whatever the order of the inserted digits), because 92 is divisible by 4.
  • If the digit root is 9, then the number is divisible by 9. The digit sum is 3 + a + 1 + b + 4 + c + 0 + d + 9 + 2. Since the order of a + b + c + d does not matter then it is always equal to 26. This makes the digit sum 3 + 1 + 4 + 0 + 9 + 2 + 26 which is equal to 45. The digit root is now 4 + 5, which is equal to 9, thus meaning that the number is divisible by 9, no matter what the order of 5, 6, 7, 8.
  • Now, using the divisibility test for 11:
    2 - 9 + d - 0 + c - 4 + b - 1 + a - 3 = a + b + c + d - 15, and since in any order, a + b + c + d = 26 this is equal to 26 - 15 = 11 . This means that the number must be divisible by 11, no matter what the order of the digits 5, 6, 7, 8.
  • Finally, since the number is divisible by 4, 9 and 11, no matter what the order of the inserted digits, then it must always be divisible by 396! This means that the probability that the answer is a multiple of 396 is equal to 1.

An alternative solution to this was to try all the numbers one at a time. This was tried successfully by Lucy and Sarah from Archbishop Sancroft High School.