Factor list
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
Age
14 to 16
Challenge level
Problem
When Tina chose a number N and wrote down all of its factors, apart from $1$ and N, she noticed that the largest of the factors in the list was $45$ times the smallest factor in the list. How many numbers N could Tina have chosen for which this is the case?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: there are 2 possible values for $N$ ($90$ or $405$)
Factors come in pairs with product $N$:
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$f$ must be prime because factors of $f$ are also factors of $N$
Factors of $45$ are also factors of $N$ so $3$ and $5$ are factors of $N$
$f\le3$ so $f=2$ or $f=3$
So $N=45\times2^2=90$ or $N= 45\times 3^2 = 405$