You may also like

problem icon

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

problem icon

14 Divisors

What is the smallest number with exactly 14 divisors?

problem icon

Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

17s and 23s

Stage: 3 Short Challenge Level: Challenge Level:1

The two digit multiples of $17$ and $23$ are $17, 34, 51, 68, 85$ and $23, 46, 69, 92$ respectively. The second digit of a pair will be the first digit of the next pair so these pairs must follow on in a certain order, eg. $17$ must follow $51$ to give $517$.
There is only one pair that offers a choice of two options: $46$ can be followed by $68$ or $69$. Using $69$ forms a loop, $69, 92, 23, 34, 46$ (giving a repeated cycle of $5$ digits $69234$). Using $68$ leads to a dead end $68, 85, 52, 17 (68517). 17$ has no pair that can follow it.

To make a $2010$ digit number, there must be a large number of loops at the start. The number can either end on the loop (on any of the $5$ digits $6, 9, 2, 3, 4$) or it can end on the dead end (on any of the $4$ digits $8, 5, 1, 7$). This gives $9$ possible endings, so there must be $9$ such numbers.

This problem is taken from the UKMT Mathematical Challenges.

View the previous week's solution
View the current weekly problem