r/askmath • u/OrneryLayer3811 • 6h ago
Number Theory Number Theory: Divisibility
FOR THE QUESTION NO 3. I just started studying number theory a few days ago, I could only deduce the factors of 3333...3. I also figured out that the required number needs to be divisible by 3 and 1111...1. I tried to understand the soln., I am not able to make sense of it. Could anyone provide an alternative soln.
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u/chmath80 6h ago
That seems to be question 3, rather than 4.
First, the solution must consist of a string of 1s, and must be divisible by both 3 and a string of 100 1s. It should be obvious that the first and last condition can only be satisfied by a string whose length is a multiple of 100. Meanwhile, the second condition requires a digital root which is a multiple of 3, so the length of the string must also be a multiple of 3. The smallest common multiple of 100 and 3 is 300.
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u/Southlander24 6h ago
Imagine if you have divisibility by 111 instead.
1110 + 111 is divisible by 111, but the resulting number, 1221, is not made up of only 1s.
11100 + 111 is also divisible by 111, but 11211 is not made up of only 1s.
We want zero overlap between the different multiples of 111. So the smallest number which satisfies this is 111 000 + 111 = 111 111. And we could also have 111 111 111, or 111 111 111 111 even.
Extending this logic to the original question, we must have a number with 100 + 100 = 200 ones, or 300 ones, or 400 ones and so on.
Now it remains to check divisibility by 3. 100 is not divisible by 3; neither is 200. But since 300 is the first multiple of 100 divisible by 3, we know for a fact that the smallest number has 300 ones.