# Puzzles

## 15 December

Today's number is smallest three digit palindrome whose digits are all non-zero, and that is not divisible by any of its digits.

## 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, ...)?

## 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?

## 14 December

In July, I posted the Combining Multiples puzzle.

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).

## Combining multiples

In each of these questions, positive integers should be taken to include 0.

1. What is the largest number that cannot be written in the form \(3a+5b\), where \(a\) and \(b\) are positive integers?

2. What is the largest number that cannot be written in the form \(3a+7b\), where \(a\) and \(b\) are positive integers?

3. What is the largest number that cannot be written in the form \(10a+11b\), where \(a\) and \(b\) are positive integers?

4. Given \(n\) and \(m\), what is the largest number that cannot be written in the form \(na+mb\), where \(a\) and \(b\) are positive integers?

## Subsum

1) In a set of three integers, will there always be two integers whose sum is even?

2) How many integers must there be in a set so that there will always be three integers in the set whose sum is a multiple of 3?

3) How many integers must there be in a set so that there will always be four integers in the set whose sum is even?

4) How many integers must there be in a set so that there will always be three integers in the set whose sum is even?

## Fill in the digits

Source: Chalkdust

Can you place the digits 1 to 9 in the boxes so that the three digit numbers formed in the top, middle and bottom rows are multiples of 17, 25 and 9 (respectively); and the three digit numbers in the left, middle and right columns are multiples of 11, 16 and 12 (respectively)?

## Always a multiple?

Source: nrich

Take a two digit number. Reverse the digits and add the result to your original number. Your answer is multiple of 11.

Prove that the answer will be a multiple of 11 for any starting number.

Will this work with three digit numbers? Four digit numbers? \(n\) digit numbers?