Divisibility

When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?