# Puzzles

## 24 December

Today's number is the smallest number with exactly 28 factors (including 1 and the number itself as factors).

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If \(p\) and \(q\) are prime numbers, then the number \(p^a\times q^b\) will have \((a+1)(b+1)\) factors. This is because all its factors are
of the form \(p^\alpha\times q^\beta\), with \(\alpha=0,1,...,a\) and \(\beta=0,1,...,b\). The same idea can be used on numbers with three or more prime factors; in general the number \(p_1^{a_1}\times...\times p_n^{a_n}\) has \((a_1+1)\times...\times(a_n+1)\) factors.

28 can be written as: 28, 14×2, 7×4, or 7×2×2. Therefore the following numbers have 28 factors:

$$2^{27},\quad
2^{13}\times3^1,\quad
2^{6}\times3^3,\quad
2^{6}\times3^1\times5^1
$$

and any other number with 28 factors will have larger prime factors, so will be larger.

These numbers are 134217728, 24576, 1728 and 960. Therefore the smallest number with 28 factors is **960**.

## 21 December

The factors of 6 (excluding 6 itself) are 1, 2 and 3. \(1+2+3=6\), so 6 is a *perfect number*.

Today's number is the only three digit perfect number.

## 20 December

What is the largest number that cannot be written in the form \(10a+27b\), where \(a\) and \(b\) are nonnegative integers (ie \(a\) and \(b\) can be 0, 1, 2, 3, ...)?

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Any number can be written as \(10a+27b\) with integer \(a\) and \(b\), since \(1=3\times27-8\times10\).
So the problem may be thought of as asking when one of \(a\) and \(b\) must be negative.

Given one way of writing a number, you can get the others by shifting by 14*29. For example,

$$219 = 10\times30 - 27\times3$$ $$= 10 + 10\times29 - 27\times3 $$ $$= 1\times10 + 27\times11$$

So the question now becomes: When does this adjustment fail to eliminate negative numbers?

This is when you are at what Pedro calls "limit coefficients":

$$10\times(-1) + 27\times13 = 10\times28 + 27\times(-1) = 233$$

So the answer is 233.

#### Extension

Let \(n\) and \(m\) be integers. What is the largest number that cannot be written in the form \(na+mb\), where \(a\) and \(b\) are nonnegative integers?

## Elastic numbers

*Throughout this puzzle, expressions like \(AB\) will represent the digits of a number, not \(A\) multiplied by \(B\).*

A two-digit number \(AB\) is called *elastic* if:

- \(A\) and \(B\) are both non-zero.
- The numbers \(A0B\), \(A00B\), \(A000B\), ... are all divisible by \(AB\).

There are three elastic numbers. Can you find them?

#### Show answer & extension

#### Hide answer & extension

15, 18 and 45 are elastic.

15's factors are 5 and 3. 105, 1005, 10005, etc will all be multiples of 5 (because they end in 5) and multiples of 3 (as their digits add to 6). Hence they are all multiples of 15.

Similarly, 108, 1008, 10008, etc are all multiples of 9 (adding digits) and 2 (they are even), so they are multiples of 18; and 405, 4005, 40005, etc are all multiples of 9 (adding digits) and 5 (last digits are 5), so they are multiples of 45.

#### Extension

How many elastic numbers are there in other bases?

## 16 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10.
Today's number is the largest number than can be made from the digits in red boxes.

| × | | × | | = 6 |

× | | × | | × | |

| × | | × | | = 180 |

× | | × | | × | |

| × | | × | | = 336 |

= 32 | | = 70 | | = 162 | |

## 14 December

Today's number is the largest number that cannot be written in the form \(27a+17b\), where \(a\) and \(b\) are positive integers (or 0).

## 8 December

Today's number is the second smallest number that can be written as
a×b×c×d×e×f×g×h×i, where
a,b,...,i are all integers greater than 1.

## 5 December

Today's number is the number of ways that 35 can be written as the sum of distinct numbers, with none of the numbers in the sum being divisible by 9.

Clarification: By "numbers", I mean (strictly) positive integers. The sum of the same numbers in a different order is counted as the same sum: eg. 1+34 and 34+1 are not different sums.
The trivial sum consisting of just the number 35 counts as a sum.