To determine if \( x \) must be a multiple of 12 when \( 7x \) is a multiple of 210, we start with the prime factorization of 210, which is \( 2 \cdot 3 \cdot 5 \cdot 7 \). Since \( 7x = 210n \) for some integer \( n \), dividing both sides by 7 gives \( x = 2 \cdot 3 \cdot 5 \cdot n \). This shows that \( x \) is a multiple of 30, not necessarily 12, as 30 does not include the factor of 4 required for multiples of 12.
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MarkFL said:I would begin by finding the prime factorization of 210, which is:
$$210=2\cdot3\cdot5\cdot7$$
Now, if $7x$ is a multiple of 210, then we may write:
$$7x=210n=2\cdot3\cdot5\cdot7\cdot n$$
Now dividing through by 7, we find:
$$x=2\cdot3\cdot5\cdot n$$
What can you conclude from this?