14 Divisors

What is the smallest number with exactly 14 divisors?

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.

Oh! Hidden Inside?

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

Differences

Stage: 3 Challenge Level:

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 $6$?