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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:
  1. \(A\) and \(B\) are both non-zero.
  2. The numbers \(A0B\), \(A00B\), \(A000B\), ... are all divisible by \(AB\).
There are three elastic numbers. Can you find them?

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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?

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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?

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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)?

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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?

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© Matthew Scroggs 2017