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14 Divisors

What is the smallest number with exactly 14 divisors?

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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.

Differences

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


Can you guarantee that, for any three whole numbers you choose, the product of their differences will always be an even number?

For example:
If your three whole numbers are $7$ , $4$ and $12$, the differences are
$7-4 = 3$
$12-7 = 5$
$12-4 = 8$.

The product of the differences is $3\times 5\times 8 = 120$ (an even number)

Start with other sets of three whole numbers. Is the product of their differences always even?

If so, can you explain why?
If not, can you find a counter-example?

AND

Can you guarantee that, for any four whole numbers you choose, the product of their differences will always be a multiple of three?

For example:
If your four whole numbers are $7$ , $4$, $12$ and $6$, the differences are
$7-4 = 3$
$12-7 = 5$
$7-6 = 1$
$12-4 = 8$
$6-4 = 2$
$12-6 = 6$

The product of the differences is $3\times 5\times 1\times 8\times 2\times 6 = 1440$ (a multiple of three)

Start with other sets of four whole numbers. Is the product of their differences always a multiple of three?
If so, can you explain why?
If not, can you find a counter-example?

In the example above, the product of the differences, $1440$, is also a multiple of $4$, $5$ and $6$.

Is the product of the differences of four numbers always divisible by $4$, $5$ and $6$?