Let $n$ be the least positive integer such that 168 is a factor of $1134^n$. If m is the least positive integer such that $1134^{n}$. is a factor of $168^m$, then $m + n$ equals

We need to find two values and then add them together.

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We need to find the prime factorizations of both numbers to determine divisibility.

For 168:

$168 = 2^3 \times 3^1 \times 7^1$

For 1134:

$1134 = 2^1 \times 3^4 \times 7^1$

When dealing with factors and multiples, prime factorizations help us see exactly what "ingredients" each number has, making it easy to determine divisibility.

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We need 168 to be a factor of $1134^n$. In other words, $1134^n$ must be divisible by 168.

When we raise 1134 to the power n:

$1134^n = (2^1 \times 3^4 \times 7^1)^n = 2^n \times 3^{4n} \times 7^n$

For 168 to divide $1134^n$, we need $1134^n$ to have at least as many of each prime factor as 168 has.

Comparing the prime factors:

168 needs $2^3$, so we need $2^n \geq 2^3$, which means $n \geq 3$

168 needs $3^1$, so we need $3^{4n} \geq 3^1$, which means $4n \geq 1$, so $n \geq \dfrac{1}{4}$

168 needs $7^1$, so we need $7^n \geq 7^1$, which means $n \geq 1$

The most restrictive condition is $n \geq 3$, so $n = 3$.

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Now we need $1134^n = 1134^3$ to be a factor of $168^m$.

We have: $1134^3 = 2^3 \times 3^{12} \times 7^3$

When we raise 168 to the power m:

$168^m = (2^3 \times 3^1 \times 7^1)^m = 2^{3m} \times 3^m \times 7^m$

For $1134^3$ to divide $168^m$, we need $168^m$ to have at least as many of each prime factor as $1134^3$ has.

Comparing the prime factors:

$1134^3$ needs $2^3$, so we need $2^{3m} \geq 2^3$, which means $3m \geq 3$, so $m \geq 1$

$1134^3$ needs $3^{12}$, so we need $3^m \geq 3^{12}$, which means $m \geq 12$

$1134^3$ needs $7^3$, so we need $7^m \geq 7^3$, which means $m \geq 3$

The most restrictive condition is $m \geq 12$, so $m = 12$.

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$m + n = 12 + 3 = 15$

When dealing with divisibility problems involving powers, always compare the prime factorizations. The number with fewer factors of any prime will determine the minimum power needed.