A four-digit number, n, is written as ‘ABCD’ where A, B, C and D are all different odd digits. It is divisible by each of A, B,C and D.
Find all the possible numbers for n.
All the possible numbers for n are 1395, 1935, 3195, 3915, 9135, 9315.
‘ABCD’ is a number composed of four out of five digits 1, 3, 5, 7, 9. Consider two cases:Case 1. 9 is not one of the digits A,B, C, D.
Number ‘ABCD’ is a multiple of 3, so A + B + C + D is also a multiple of 3. But A + B + C + D = 1 + 3 + 5 + 7 = 16, which is not a multipleof 3. This is a contradiction, so there are no solutions in this case.
Case 2. 9 is one of the digits A, B, C, D.
Number ‘ABCD’ is a multiple of 9, so A + B + C + D is also a multiple of 9.
Let x be the digit not used in ‘ABCD’.Then A + B + C + D = 1 + 3 + 5 + 7 + 9 − x = 25 − x
and this must be a multiple of 9. Hence x = 7.
This means 1, 3, 5, 9 are the digits used, so ‘ABCD’ is a multiple of 5. Hence D = 5. We can rearrange 1, 3, 9 as A, B, C in six different ways.
All the possible numbers for n are 1395, 1935, 3195, 3915, 9135,9315. All of them are divisible by 1, 3 and 9 (as their sum of digits is always 18), and also by 5 (as they all end in a 5).
Be up-to-date with our recent updates, new problems and answers!
Ask about our courses and offerings, and we will help you choose what works best for you.
