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.

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.

You might enjoy playing The Remainders Game next