Factors and multiples

Can you place the numbers from 1 to 10 in the grid?

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?

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?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.