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3 integers: x, y, z if there is a multiple of 7?

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mihai9 Junior | Next Rank: 30 Posts Default Avatar
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3 integers: x, y, z if there is a multiple of 7?

Post Thu Nov 09, 2017 1:26 pm
3 integers: x, y, z if there is a multiple of 7?
1: x + y + z is a multiple of 7
2: xyz is a multiple of 7

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Post Thu Nov 09, 2017 7:02 pm
Let's rephrase the prompt as

Quote:
x, y, and z are all integers. Is at least one of them a multiple of 7?
S1::

x + y + z = 7 * some integer

We could have 1 + 2 + 4, or we could have 7 + 14 + 21; NOT SUFFICIENT.

S2::

xyz = 7 * some integer

Since each side consists entirely of integers, each side must have the same prime factorization. Since 7 appears in the prime factorization of the right side, it must also appear in the prime factorization of the left side. That means that at least one of x, y, and z contains a factor of 7 in its prime factorization, so one of them must divide by 7.

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