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.
Correct solutions were received from Andrei of School 205
Bucharest (whose solution forms the basis of the one below), Mary
of Birchwood High and Chen from The Chinese High School,
If two numbers are both divisible by 7, then their sum will
always be divisible by 7.
Any 3-digit number could be written as:
abc = 100a + 10b + c
We must prove that if 2a + 3b + c is divisible by 7, then
abc is also divisible by 7.
Subtracting (2a + 3b + c) [which is divisible by 7] from 100a +
10b + c [the value of the number abc], gives:
98a + 7b = 7 (14a + b)
This number is always divisible by 7.
This means that abc is divisible
by 7 when (2a + 3b + c) is divisible by 7.