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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The other three digit numbers that have 3 and 5 as factors and no other prime factors are:
- 31×53 = 375
- 32×52 = 225
- 33×51 = 135
- 33×52 = 675
- 34×51 = 405
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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In order for the number of odd factors of the number to be odd, it must be an odd square number or a power of two times an odd square number.
128 is 1 (an odd square) times 128 (a power of two) and no smaller three digit number is of this form.
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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If the numbers \(p_1\), \(p_2\), ..., \(p_n\) are odd prime numbers and \(i_1\), \(i_2\), ..., \(i_n\) are positive integers, then the number \(p_1^{i_1}p_2^{i_2}...p_n^{i_n}\)
has \((i_1+1)(i_2+1)...(i_n+1)\) odd factors.
If \((i_1+1)(i_2+1)...(i_n+1)=16\) the the possible values for the \(i\)s are:
- 15
- 7 and 1
- 3 and 3
- 3, 1 and 1
- 1, 1, 1 and 1
These options lead to:
- 315 = 14348907
- 37×51 = 10925
- 33×53 = 3375
- 33×51×71 = 945
- 31×51×71×111 = 1155
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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The factors will all be of the form \(2^a\times5^b\times7^c\times11^d\), where
\(0\leqslant a\leqslant26\),
\(0\leqslant b\leqslant1\),
\(0\leqslant c\leqslant5\), and
\(0\leqslant d\leqslant2\). There are 27 choices for \(a\), 2 for \(b\), 6 for \(c\), and 3 for \(d\) giving a total of 27×2×6×3 = 972 factors.
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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Square numbers are the only numbers where the geometric mean of their factors is equal to a whole number,
and in each case the geometric mean will be the square root. Therefore the only number
where the geometric mean of its factors is 13 is 169.
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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If \(p_1^{a_1}\times p_2^{a_2}\times\dots\times p_n^{a_n}\) is the prime factorisation of a number, then the number has \((a_1+1)(a_2+1)\dots(a_n+1)\) factors.
The prime factorisation of 120 is \(2^3\times3\times5\). The next smallest number with 16 factors must be one of:
- \(2\times3^3\times5=270\)
- \(2^3\times3\times7=168\)
- \(2^3\times3^3=216\)
The smallest of these is 168.
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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8×7×5×3=840. This is also a multiple of 4, 6, 2, and 1. If anything is removed from the product it will no longer be a multiple of all the numbers.
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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The first division tells that 12345 is 205 more than a mutliple of today's number, and so today's numwber is a factor of 12140.
The second division tells that 6789 is 112 more than a mutliple of today's number, and so today's number is a factor of 6677.
The only numbers that are factors of both 12240 and 6677 are 1 and 607.
