Question

If 12 divides ab313ab then smallest value of a+b is

Answer 1

If 12 divided ab313ab, then the sum of digits of ab313ab should be divided by 3, and the two digit formed by joining a and b, i. e., ab, should be divided by 4.

2(a + b) + 3 + 1 + 3 is a multiple of 3.

Among 313, 1 is included. So ab should have the values of any terms in the sequence 1, 4, 7, 10,... but of multiple of 4 also. a + b results 1, 4, 7, 10,... also.

'a' cannot be 0 because if taken, ten lakhs digit of ab313ab will be lost and become b313ab. So 1, 4, 7 are avoided. But 'b' can be 0.

ab can be 10 but 10 is not a multiple of 4.

ab can be 13 but not divided by 4.

But ab can be 16 as it's also a multiple of 4. 1 + 6 = 7. So 7 can be the value of a + b.

Here, 10 cannot be had. So a + b can't be had 1 + 0 = 1 because only the two digit number 10 results in the sum of its digits as 1. 4 has the chance to be the smallest value of a + b. Let's check.

16 can be had. 1 + 6 = 7.

28 can be had. 2 + 8 = 10.

40 can be had. 4 + 0 = 4.

So 4 can be the value of a + b.

∴ 4 is the smallest value of a + b.

If ab = 40, then ab313ab = 4031340.

4031340 ÷ 12 = 335945.

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