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If the five digit number A679B, in base ten, is divisible by 72 then A = 3 and B = 2. The number is 36792.
This is how Jamie from Hethersett High School Norfolk explained his method.
To be divisible by 72 the number must be divisible by 8 and 9 as 8 x 9 = 72. I don't know a rule for finding if a number is divisible by 8, but I realised that I could look for a number divisible by 4. The rule for this is that if the last two digits of a number are divisible by 4 then the whole number is divisible by 4. In the nineties (as, in the question, 9 is the second from the last digit) only 92 and 96 are divisible by 4. Bearing these numbers in mind I brought the rule about a number being divisible by 9 into play. To be divisible by 9 the sum of all the digits in the number added together must be divisible by 9. These are the sums I did.
A + 6 + 7 + 9 + 2 = 24 + A. To make this up to a number divisible by 9, A would have to be 3.
A + 6 + 7 + 9 + 6 = 28 + A. To make the number divisible by 9, A would have to be 8. However when 86796 is divisible by 4 and 9 but not by 8.
You may like to try a similar problem sent by John of Madras College: ```A512B is divisible by 65, find A and B''.
Well done Sonya, Genevieve, James, and Angela of Hethersett High School, Norfolk; Claire, Kim, Joanna, Rhona, David, Elisabeth, Nicola, Colin, Ross, Myles, and Helen from Madras College, Scotland; Danielle from The Mount School York; and Chin Siang and Xinxin of Tao Nan School, Singapore for your excellent solutions.
The following solution came from Stephen.
We have been learning BBC BASIC since September and have just learnt the for-next loop. We used this to write a programme, shown below, to solve this problem.
10 FOR A = 1 TO 9
20 FOR B = 0 TO 9
30 IF (10000*A + 6790 + B)/72 = INT((10000*A + 6790 + B)/72) THEN PRINT A,B
40 NEXT B
50 NEXT A
So A = 3 and B = 2.
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
Can you explain the strategy for winning this game with any target?
Is there an efficient way to work out how many factors a large number has?