Brainsteem Quickfire Q.7
The Brainsteem Quickfire questions should (hopefully) be a bit quicker than the standard questions.
As always, these should be done by hand without any computational assistance! The point is to discover something interesting and something new!
Question 7
Let N be a decimal number using only the numerals 0 and 1.
Find the smallest number of digits of N such that N is divisible by 56.
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This question is far easier to solve when broken down based on the prime factorization of 56, which is 23 x 7.
First we look at 23 = 8. We need an even number, therefore the number we want must end in 0. The smallest decimal number consisting of only 1s and 0s that is also divisible by 8 is 1000.
Now for 7, we need an odd number so it must end in 1. The smallest decimal number consisting of only 1s and 0s that is also divisible by 7 is 1001.
N = 1000 x 1001 = 1001000, which is definitely divisible by 56.
The number of digits required is 7.
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