Prime Factorization - Is Number a Multiple of Both - From Values as Factors (Level 3)

This math topic focuses on advanced concepts of prime factorization and checking whether one number is a multiple of others, using their prime factorized forms. It involves determining if a primary number is a multiple of two other numbers by analyzing their factorized structures, as presented in the format of prime factors raised to appropriate exponents. Additionally, this topic falls under a broader unit on factoring and finding the lowest common multiple, targeting an advanced understanding of these concepts. Each problem presents a situation to decide if one number is a multiple of the others provided, with a choice of 'Yes' or 'No' as answers.

Is Number a Multiple of Both - From Values as Factors

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Is 4802 a multiple of both 1078 and 686?

4802=2741078=27211686=273is 4802 a multiple of 1078 and 686?\begin{align*}4802 &= 2 \cdot 7^4\\\\[-0.5em]1078 &= 2 \cdot 7^2 \cdot 11\\[-0.5em]686 &= 2 \cdot 7^3\end{align*}\\\\ \textsf{is }4802\textsf{ a multiple of }\\1078\textsf{ and }686?