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Puzzles

24 December

3 and 5 are both factors of 2025, and 3 and 5 are the only two prime numbers that are factors of 2025.
What is the largest three-digit number that has both 3 and 5 as factors and no other prime numbers as factors?

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15 December

The odd factors of 2025 are 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675 and 2025. There are 15 of these factors and 15 is itself an odd factor of 2025.
What is the smallest three-digit number whose number of odd factors is itself an odd factor of the number?

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10 December

2025 is the smallest number with exactly 15 odd factors.
What is the smallest number with exactly 16 odd factors?

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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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