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.
List any 3 numbers.
It is always possible to find a subset of adjacent numbers that add up to a multiple of 3 (that is either one, two or three numbers that are next to each other). For example:
Can you explain why and prove it?
What happens if you write a list of 4 numbers? Is it always possible to find a subset of adjacent numbers that add up to a multiple of 4? Can you explain why and prove it?
What happens if you write a long list of numbers (say n numbers)? Is it always possible to find a subset of adjacent numbers that add up to a multiple of $n$? Can you explain why and prove it?