Charlie is thinking of a number.

His number is both a multiple of $5$ and a multiple of $6$.

**What could his number be?**

Alison is thinking of a number.

Her number is a multiple of $4$, $5$ and $6$.

**What could her number be?**

Charlie is thinking of a number that is $1$ more than a multiple of $7$.

Alison is thinking of a number that is $1$ more than a multiple of $4$.

**Could they be thinking of the same number?**

Charlie is thinking of a number that is $3$ more than a multiple of $5$.

Alison is thinking of a number that is $8$ more than a multiple of $10$.

**Could they be thinking of the same number?**

Charlie is thinking of a number that is $3$ more than a multiple of $6$.

Alison is thinking of a number that is $2$ more than a multiple of $4$.

**Could they be thinking of the same number?**

The interactivity below can be used to check your answers. It allows you to choose a divisor and then select numbers in one of the columns. Here are a couple of examples of how it can be used.

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Here is a set of cards

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.

Full Screen Version

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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.