Pairing up
The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?
Age
11 to 14
Challenge level
Problem
The numbers $72, 8, 24, 10, 5, 45, 36, 15$ are grouped in pairs so that each pair has the same product. Which number is paired with $10$?
If you liked this question, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $36$
Using logic
As the product of each pair has the same value, this value must be product of the smallest and largest numbers, that is $5\times72$.
So the number which is paired with $10$ is $5\times72\div10$, that is $36$.
Using prime factors
Only four have a factor of $5$ so each pair must contain one of these
Image
$5$ has no factors of $2$ or $3$, so pairs with $72$, which has the highest powers of $2$ and $3$
Image
To get $2^3$ and $3^2$, $10$ pairs with $36$
Image
The others can also be paired up:
Image
