Challenge Level

I'm thinking of a number.

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

**What could my number be?
What else could it be?
What is the smallest number it could be?**

I'm thinking of a number.

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

**What could my number be?
What else could it be?
What is the smallest number it could be?**

The Number Sieve below can be used to explore questions like the ones above and many more. Why not experiment and see what you can discover?

**Here are some more questions you might like to consider:**

I'm thinking of a number that is $1$ more than a multiple of $7$.

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

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

I'm thinking of a number that is $3$ more than a multiple of $5$.

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

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

I'm thinking of a number that is $3$ more than a multiple of $6$.

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

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

Here's a challenging extension:

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 a 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?

**Can you find the smallest number that satisfies this condition?**

*If you haven't already explored The Remainders Game, why not take a look now?*