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
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.
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 $N= 45\times 3^2 = 405$