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Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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Skeleton

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

Remainders

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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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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

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