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

This math topic focuses on advanced skills related to prime factorization, particularly assessing if one number is a multiple of other numbers using variable factors and exponents. Participants determine if specific values of a variable (represented as products of primes and their powers) are multiples of given numbers. The problems employ factorization and concepts of lowest common multiples to analyze mathematical relationships and verify multiple conditions. Additionally, this includes practice in translating between numerical expressions and contextualizing mathematical statements.

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Is Number a Multiple of Both - From Variables as Factors

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Is n a multiple of both 60 and 250?

\begin{align*}n &= 2^2 \cdot 5^3\\\\[-0.5em]60 &= 2^2 \cdot 3 \cdot 5\\[-0.5em]250 &= 2 \cdot 5^3\end{align*}\\\\ \textsf{is }n\textsf{ a multiple of }\\60\textsf{ and }250?

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Prime Factorization - Is Number a Multiple of Both - From Variables as Factors Worksheet

Math worksheet on 'Prime Factorization - Is Number a Multiple of Both - From Variables as Factors (Level 3)'. Part of a broader unit on 'Factoring and Lowest Common Multiple - Advanced' Learn online: app.mobius.academy/math/units/factoring_and_lowest_common_multiple_advanced/

$A LaTex expression showing \begin{align*}x &= 3 to the power of 3 times 5 times 7\\\\[-0.5em]459 &= 3 to the power of 3 times 17\\[-0.5em]315 &= 3 to the power of 2 times 5 times 7\end{align*}\\\\ \textsf{is }x\textsf{ a multiple of }\\459\textsf{ and }315?$

Is x a multiple of both 459 and 315?

Yes

$A LaTex expression showing \begin{align*}m &= 2 to the power of 3 times 7 to the power of 2 \\\\[-0.5em]56 &= 2 to the power of 3 times 7\\[-0.5em]196 &= 2 to the power of 2 times 7 to the power of 2 \end{align*}\\\\ \textsf{is }m\textsf{ a multiple of }\\56\textsf{ and }196?$

Is m a multiple of both 56 and 196?

Yes

$A LaTex expression showing \begin{align*}y &= 2 times 5 to the power of 2 times 7 to the power of 2 \\\\[-0.5em]490 &= 2 times 5 times 7 to the power of 2 \\[-0.5em]350 &= 2 times 5 to the power of 2 times 7\end{align*}\\\\ \textsf{is }y\textsf{ a multiple of }\\490\textsf{ and }350?$

Is y a multiple of both 490 and 350?

Yes

$A LaTex expression showing \begin{align*}y &= 2 times 3 to the power of 2 times 7 to the power of 2 \\\\[-0.5em]210 &= 2 times 3 times 5 times 7\\[-0.5em]294 &= 2 times 3 times 7 to the power of 2 \end{align*}\\\\ \textsf{is }y\textsf{ a multiple of }\\210\textsf{ and }294?$

Is y a multiple of both 210 and 294?

Yes

$A LaTex expression showing \begin{align*}p &= 2 times 3 to the power of 3 times 7\\\\[-0.5em]90 &= 2 times 3 to the power of 2 times 5\\[-0.5em]126 &= 2 times 3 to the power of 2 times 7\end{align*}\\\\ \textsf{is }p\textsf{ a multiple of }\\90\textsf{ and }126?$

Is p a multiple of both 90 and 126?

Yes

$A LaTex expression showing \begin{align*}n &= 2 to the power of 3 times 5 times 7\\\\[-0.5em]140 &= 2 to the power of 2 times 5 times 7\\[-0.5em]40 &= 2 to the power of 3 times 5\end{align*}\\\\ \textsf{is }n\textsf{ a multiple of }\\140\textsf{ and }40?$

Is n a multiple of both 140 and 40?

Yes

$A LaTex expression showing \begin{align*}r &= 2 times 5 times 7 to the power of 3 \\\\[-0.5em]490 &= 2 times 5 times 7 to the power of 2 \\[-0.5em]686 &= 2 times 7 to the power of 3 \end{align*}\\\\ \textsf{is }r\textsf{ a multiple of }\\490\textsf{ and }686?$

Is r a multiple of both 490 and 686?

Yes

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

Practice:

Is Number a Multiple of Both - From Variables as Factors

Prime Factorization - Is Number a Multiple of Both - From Variables as Factors Worksheet