What is the smallest number with exactly 14 divisors?
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.
Find the number which has 8 divisors, such that the product of the divisors is 331776.
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 $6$?