# Puzzles

## 22 December

In base 2, 1/24 is
0.0000101010101010101010101010...

In base 3, 1/24 is
0.0010101010101010101010101010...

In base 4, 1/24 is
0.0022222222222222222222222222...

In base 5, 1/24 is
0.0101010101010101010101010101...

In base 6, 1/24 is
0.013.

Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.

Today's number is the smallest base in which 1/10890 has a finite number of digits.

Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).

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If 1/10890 has a finite number of digits in base \(b\), then \(b^n/10890\) is an integer for some integer \(n\). In order for this to be
possible \(b\) must be a mutiple of each prime factor of 10890: the smallest such \(b\) will be when the exponent of each prime factor is 1.

The prime factorisation of 10890 is 2×3^{3}×5×11^{2}. Therefore the smallest base in which 1/10890 has a
finite number of digits is 2×3×5×11=**330**.

## 121

Find a number base other than 10 in which 121 is a perfect square.

## Adding bases

Let \(a_b\) denote \(a\) in base \(b\).

Find bases \(A\), \(B\) and \(C\) less than 10 such that \(12_A+34_B=56_C\).

## Reverse bases again

Find three digits \(a\), \(b\) and \(c\) such that \(abc\) in base 10 is equal to \(cba\) in base 9?

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445 in base 10 is equal to 544 in base 7.

#### Extension

Find another pair of bases \(A\) and \(B\) so that there exist digits \(d\), \(e\) and \(f\) such that \(def\) in base \(A\) is equal to \(fed\) in base \(B\)?

## Reverse bases

Find two digits \(a\) and \(b\) such that \(ab\) in base 10 is equal to \(ba\) in base 4.

Find two digits \(c\) and \(d\) such that \(cd\) in base 10 is equal to \(dc\) in base 7.

Find two digits \(e\) and \(f\) such that \(ef\) in base 9 is equal to \(fe\) in base 5.

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If \(ab\) in base 10 is equal to \(ba\) in base 4, then \(10a+b=4b+a\).

So, \(9a=3b\).

\(a\) and \(b\) must both be less than 4, as they are digits used in base 4, so \(a=1\) and \(b=3\).

So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:

- 23 in base 10 is equal to 32 in base 7.
- 46 in base 10 is equal to 64 in base 7.
- 12 in base 9 is equal to 21 in base 5.
- 24 in base 9 is equal to 42 in base 5.

#### Extension

For which pairs of bases \(A\) and \(B\) can you find two digits \(g\) and \(h\) such that \(gh\) in base \(A\) is equal to \(hg\) in base \(B\)?