Here is a chance to explore some properties of numbers and then have a go at some questions.

The interactivity below allows you to choose a divisor and then select numbers in one of the columns.

Example 1:

Divide by 4 and select all the numbers in the right-hand column - they should all turn red.

Now divide by 5 and select all the numbers in the right-hand column - they should all turn yellow, but some will turn orange.

What is special about the numbers that turn orange?

Now divide by 3 and select all the numbers in the right-hand column - most should turn blue, but one will turn black.

What is special about the number that turns black?

What is special about the numbers that turn green and purple?

Example 2: (you will need to clear your previous work)

Find the numbers that have a remainder of 2 when divided by 5 - you'll need to divide by 5 and select the numbers in the second column.

Now select the numbers that have a remainder of 1 when divided by 2 (the odd numbers).

What is special about the numbers that turned orange this time?

Try a few examples of your own and try to predict what will happen in each case.

Now try out the problem generator below. When you click "Start" the computer will select at random an integer between 1 and 100. Can you identify the chosen number?

You can use the interactivity above to help you, but eventually, try to identify the numbers without the aid of the interactivity.

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One final question:

We know that

When 59 is divided by 5, the remainder is 4

When 59 is divided by 4, the remainder is 3

When 59 is divided by 3, the remainder is 2

When 59 is divided by 2, the remainder is 1

When 59 is divided by 4, the remainder is 3

When 59 is divided by 3, the remainder is 2

When 59 is divided by 2, the remainder is 1

Can you find the smallest number with the property that when it is divided by each of the numbers 2 to 10, the remainder is always one less than the number it is has been divided by? Don't forget to explain your reasoning.