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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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Cube multiples

Six different (strictly) positive integers are written on the faces of a cube. The sum of the numbers on any two adjacent faces is a multiple of 6.
What is the smallest possible sum of the six numbers?

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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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23 December

In the song The Twelve Days of Christmas, how many presents have been given after 8 days?

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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