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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?

Show Answer & Extension

18 December

The smallest number whose sum of digits is 25.

10 December

The smallest number that is equal to the sum of its digits multiplied by ten more than the sum of its digits.

6 December

When you add up the digits of a number, the result is called the digital sum.
How many different digital sums do the numbers from 1 to 1091 have?*
* There was a mistake in this question (it previously said 1092). If you answered the typo'd question right, your answer should automatically correct itself to 9 less than it was...

1 December

One of the digits of today's number was removed to leave a two digit number. This two digit number was added to today's number. The result was 619.

Santa

Each of the letters D, A, Y, S, N, T, B, R and E represents a different non-zero digit. The following sum is true:
$$ \begin{array}{cccccc} D&A&D&D&Y\\ B&E&A&R&D&+\\ \hline S&A&N&T&A \end{array} $$
This has a unique solution, but I haven't found a way to find the solution without brute force. This less insightful sum is also true with the same values of the letters (and should allow you to find the values of the letters using logic alone):
$$ \begin{array}{ccccc} R&A&T&S\\ N&E&R&D&+\\ \hline S&A&N&E \end{array} $$

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© Matthew Scroggs 2017