UKMT Special (Problem 1414)

No digit of the positive integer NN is prime. However, all the the single-digit primes divide NN exactly.

What is the smallest such integer NN?

[UKMT Hamilton Olympiad 20162016, H22]

Note by Yajat Shamji
8 months, 2 weeks ago

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You have until next Monday, 3:003:00pm!

Yajat Shamji - 8 months, 2 weeks ago

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Step 1) The first statement implies that the possible digits of N are 0, 1, 4, 6, 8, and 9.

Step 2) The first statement implies that N is divisible by 2, 3, 5, and 7.

Step 3) Because 2 and 5 divide N exactly, 10 must also divide N. This implies that the last digit of N is 0.

Step 4) Now, lets try listing some positive integer that divisible by 3, 7, and 10.

   210, 420, 630, 840, ...

Notice that 840 is the smallest positive integer that satisfy the statement.

PS : I notice that the third step is not necessary, but it might give you insight on other problem.

Quest Keeper - 8 months, 2 weeks ago

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Correct, but slightly different method.

Yajat Shamji - 8 months, 2 weeks ago

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