Factorising numbers $1 , 2 , 3 , 4 , \dots$ until you find the first one with 14 divisors will give the answer here, which is $192$, but there is a better way! You could write a simple computer program ... but there is a more mathematical way!
Every whole number can be factorised into prime factors and here we are looking for a number like $(2^a)(3^b)(5^c)$ etc. Note that $2^a$ has $(1+a)$ factors so $2^{13}$ has 14 factors but could a number like $(2^a)(3^b)(5^c)$ be smaller?
Consider $(2^a)(3^b)$. This has $(1+a)(1+b)$ factors and so we want $(1+a)(1+b)=14$. Since $14=2 \times 7$ this must give $a=6$ and $b=1$ if we want to make the number as small as possible, and $2^{6}3^{1} = 192$ which is a lot smaller than $2^{13}$.
It remains to prove that it is impossible to find a smaller number which has more prime factors, say $(2^a)(3^b)(5^c)$, but this can't be done because 14 itself has only two factors. If we look for $a$, $b$ and $c$ such that $(1+a)(1+b)(1+c) = 14$ we find that $a$ or $b$ or $c$ must be zero.
A simple BASIC program might be something like this:
10 X=0
20 REPEAT
30 X=X+1
40 D=0
50 FOR Y = 1 TO X
60 R=X/Y-INT(X/Y)
70 IF R=0 THEN D=D+1
80 NEXT Y
90 UNTIL D=14
100 PRINT "Smallest number with exactly 14 divisors =";X
The above program is in
BBC BASIC
. If you are using
QBASIC
you will need to use a DO UNTIL LOOP in place of the REPEAT......UNTIL loop.
Correct solutions to this problem were submitted by
Rachel Took , Mary Egan and Rose Paul , West Flegg Middle School
Ruth Bradman , Lynn Grove High School
Hannah Stow
,
Lucy Hannah
,
Lawrence Tardelier
,
Sarah Gray
and
Sarah Hannah
, Archbishop Sancroft High School
Wymondham High School