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## 'Differences' printed from http://nrich.maths.org/

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 $6$?