Find a great variety of ways of asking questions which make 8.
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
The number $12 = 2^2 \times 3$
This is how Graeme Gilpin from Newcastle Royal Grammar School
arrived at the answer:
I worked out what was the lowest number that had 12 factors and
came to 60. I then tried numbers over 200 that were multiples of 2,
3 and 5, e.g. 210. I realised after about an hour that no number
under 1000 had exactly 36 factors. I then started again with the 2,
3 and 5 sequence until I got to a number that worked, which is
1260. All in all it took me about one and a half hours to work out
The factors are : 1260, 1, 630, 2, 420, 3, 315, 4, 252, 5, 210,
6, 180, 7, 140, 9, 126, 10, 105, 12, 90, 14, 84, 15, 70, 18, 63,
20, 60, 21, 45, 28, 42, 30, 36, 35.
Ken Nisbet from Madras College, St Andrew's wrote:
"36 Exactly was done by the whole class 3XP over several days and
developed with ideas from the class. The crucial 'formula' was
discovered by Dorothy Winn. Needless to say it was thereafter
referred to as the 'Winning Formula'! I have sent what I considered
the best four write-ups although each student in the class produced
a write up for this question."
A systematic approach, which works in general, depends on prime
factorisation as class 3XP discovered. Thank you to Claire
Kruithof, Catherine Aitken, and Joe Neilson for your splendid
write-ups. Well done 3XP!
Here is Dorothy's write-up:
"We were asked to find the smallest positive integer with exactly
36 factors. I started to look for numbers using trial and error. I
took smaller numbers with lots of factors, like 36, and multiplied
them by other numbers with many factors to see if I could get a
larger number with even more factors. I didn't get very far with
that though and I began to make a table which had all the numbers
up to 40, their factors, prime factors and number of factors. I
colour coded it to make it easier to see numbers which had the same
number of factors.
We found that there is a connection between the number of
factors and the prime factorisation. I found that if you add one to
each of the indices and multiply these numbers together you get the
total number of factors."
[Editor's note: For example, $(2,1)$
You can do it backwards by finding the smallest numbers that
multiply together to get 36, subtracting one from each of them and
using them as indices. The smallest numbers that multiply together
to get 36 are (2, 2, 3, 3). If you subtract one from each of them
you get (1, 1, 2, 2). If you use these as indices for primes the
smallest number you can get is $7^1 \times 5^1 \times 3^2 \times
2^2$ which comes to 1260. I have put the numbers in that order (7,
5, 3, 2) because the answer will be less if take lower powers of
the higher numbers and higher powers of the lower numbers, squaring
2 and 3 and not 5 and 7.
You can also get 36 factors with (4,3,3). This
gives$2^3 \times 3^2 \times 5^2 = 1800$ a number
greater than 1260, or (2, 2, 9) which gives you$2^8
\times 3 \times 5 = 3840$ , much greater than 1260.
[ Editors note: others wrote out all the index patterns for
36 factors, and also all the factors of 1260 written as products of
the prime numbers 2, 3, 5 and 7.]