A number theory problem by Daniel Patten
Number Theory
Level
2
Which of the following numbers is a multiple of 36?
1896
1782
1236
1944
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1 solution
An easy way to know if a number is a multiple of 36 is to check that is both a multiple of 4, and a multiple of 9. 36 is the lowest common multiple of these two numbers, and checking them is quite straightforward really. For 4, one must only check the last two digits: obviously, the last one must be even (0, 2, 4, 6, 8), but it's a good idea to split the even numbers into two groups. If the last digit is a 0, 4 or an 8, and the second to last one is also even, then it is a multiple of 4. If the last digit is a 2 or a 6, and is preceded by an odd number, then it's a multiple of 4. For this reason, we can rule out 1782. As for 9, all we have to do is add up all the digits in the number, and if their sum total is 9; or another multiple of 9, then the number is divisible by 9. Of the three remaining answers, only one satisfies this rule; 1944. 1 + 9 + 4 + 4 = 18. The sum total of digits for 1896 and 1236 is 24 and 12 respectively, therefore they are eligible.