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Puzzles

17 December

The number 40 has 8 factors: 1, 2, 4, 5, 8, 10, 20, and 40.
How many factors does the number 226×5×75×112 have?

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4 December

The geometric mean of a set of n numbers is computed by mulitplying all the numbers together, then taking the nth root.
The factors of 9 are 1, 3, and 9. The geometric mean of these factors is
$$\sqrt[3]{1\times3\times9}=\sqrt[3]{27}=3$$
What is the smallest number where the geometric mean of its factors is 13?

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19 December

120 is the smallest number with exactly 16 factors (including 1 and 120 itself).
What is the second smallest number with exactly 16 factors (including 1 and the number itself)?

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2 December

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 7, and 8?

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6 December

When 12345 is divided by today's number, the remainder is 205. When 6789 is divided by today's number, the remainder is 112.

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2 December

The number \(7n\) has 37 factors (including 1 and the number itself). How many factors does \(8n\) have?
There was a typo in this puzzle. It originally read "38 factors" when it was meant to say "37 factors".

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15 December

When talking to someone about this Advent calendar, you told them that the combination of XMAS and MATHS is GREAT. They were American, so asked you if the combination of XMAS and MATH is great; you said SURE. You asked them their name; they said SAM.
Each of the letters E, X, M, A, T, H, S, R, U, and G stands for a different digit 0 to 9. The following sums are correct:
Today's number is SAM. To help you get started, the letter T represents 4.

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15 December

There are 5 ways to make 30 by multiplying positive integers (including the trivial way):
Today's number is the number of ways of making 30030 by multiplying.

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