For distinct prime numbers p and q, two positive integers are defined as a = p^3 q^2 and b… — GMAT

For distinct prime numbers $p$ and $q$ , two positive integers are defined as $a = p^3 q^2$ and $b = p^2 q^4$ . What is the ratio of the least common multiple of $a$ and $b$ to the greatest common divisor of $a$ and $b$ ?

$p q^2$ Cevap
B
$p^3 q^4$
C
$p^2 q^2$
D
$p^5 q^6$
E
$p q$

Cevap

p q^2

The greatest common divisor of

a = p^3 q^2

and

b = p^2 q^4

p^2 q^2

by taking the minimum exponent for each prime factor. The least common multiple is

p^3 q^4

by taking the maximum exponent for each prime factor. Dividing the LCM by the GCD yields

\frac{p^3 q^4}{p^2 q^2} = p q^2

Adım Adım Çözüm

Find the prime factorization representation of the greatest common divisor (GCD) of

a

and

b

\text{GCD}(a, b) = p^{\min(3, 2)} q^{\min(2, 4)} = p^2 q^2

The GCD taking prime factorizations uses the minimum power of each common prime factor.

Find the prime factorization representation of the least common multiple (LCM) of

a

and

b

\text{LCM}(a, b) = p^{\max(3, 2)} q^{\max(2, 4)} = p^3 q^4

The LCM taking prime factorizations uses the maximum power of each prime factor.

Compute the ratio of the LCM to the GCD.

\frac{\text{LCM}(a, b)}{\text{GCD}(a, b)} = \frac{p^3 q^4}{p^2 q^2} = p^{3-2} q^{4-2} = p q^2

Dividing exponential terms with the same base requires subtracting their exponents.

Anahtar Kavram

Calculating GCD and LCM using prime factor exponent rules and evaluating their ratio

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Adım Adım Çözüm

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