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Square and cube endings

Source: UKMT 2011 Senior Kangaroo
How many positive two-digit numbers are there whose square and cube both end in the same digit?

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Digitless factor

Ted thinks of a three-digit number. He removes one of its digits to make a two-digit number.
Ted notices that his three-digit number is exactly 37 times his two-digit number. What was Ted's three-digit number?

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Backwards fours

If A, B, C, D and E are all unique digits, what values would work with the following equation?
$$ABCCDE\times 4 = EDCCBA$$

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11 December

Two more than today's number is the reverse of two times today's number.

4 December

Pick a three digit number whose digits are all different.
Sort the digits into ascending and descending order to form two new numbers. Find the difference between these numbers.
Repeat this process until the number stops changing. The final result is today's number.

1 December

Today's number is the smallest three digit number such that the sum of its digits is equal to the product of its digits.

XYZ

Which digits \(X\), \(Y\) and \(Z\) fill this sum?
$$ \begin{array}{cccc} &X&Z&Y\\ +&X&Y&Z\\ \hline &Y&Z&X \end{array} $$

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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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© Matthew Scroggs 2018