Guess the Number
Number Theory
Level
3
Find the least natural number is multiply of 84 and only has the digit 7 or 6.
The answer is 76776.
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1 solution
The question is not well written. It should say, "Find the lowest multiple of 84 that only has the digits 7 or 6."
Since any multiple of 84 must be even, the number must end in a 6. Also, since 6 is a factor of 84, the number must be divisible by 6, in other words the sum of its digits must be divisible by 3. So it could be 666, 6666, 66666, etc.
If it includes a 7, then it must include three 7s or a multiple of three 7s, since that is the only way that the digits can sum to a number divisible by 3. So the first possibilities are 67776, 76776, 77676, and 77766.