Let m and n be positive integers such that 72^2 \cdot m = n^3. Which of the following stat… — GRE General Test

Let $m$ and $n$ be positive integers such that $72^2 \cdot m = n^3$ . Which of the following statements MUST be true? Select all that apply.

$m$ is divisible by $9$ Cevap
$n$ is divisible by $36$ Cevap
C
$m$ must be a perfect square
D
$n$ is divisible by $54$
The product $m \cdot n$ is divisible by $108$ Cevap

Cevap

The statements that

m

is divisible by

9

n

is divisible by

36

, and the product

m \cdot n

is divisible by

108

MUST be true.

Analyzing prime factorizations shows that

n^3 = 2^6 \cdot 3^4 \cdot m

. For the right-hand side to be a perfect cube, the power of

3

must be elevated to at least

6

, requiring

m

to contain

3^2 = 9

as a factor. Consequently,

n^3

contains at least

2^6 \cdot 3^6 = 36^3

, forcing

n

to be a multiple of

36

. Finally, multiplying

m = 9k^3

n = 36k

yields

324k^4

, which is a multiple of

108

for all integer values of

k

Adım Adım Çözüm

Express

72^2

in terms of its prime factorization.

72 = 2^3 \cdot 3^2 \implies 72^2 = (2^3 \cdot 3^2)^2 = 2^6 \cdot 3^4

Prime factorization allows us to analyze the exponents required for

n^3

to be a perfect cube.

Determine the minimum prime factor requirements for

m

and

n

2^6 \cdot 3^4 \cdot m = n^3 \implies

exponent of

2

m

must be

\ge 0

(a multiple of 3), exponent of

3

m

must be

\ge 2

(since

4 + 2 = 6

is a multiple of 3).

Every prime factor in a perfect cube must have an exponent divisible by 3.

Establish general algebraic expressions for

m

and

n

m = 9k^3

and

n = 36k

for any positive integer

k

This captures all possible integer solutions for

m

and

n

Test each statement using the general forms

m = 9k^3

and

n = 36k

m

is divisible by

9

(

9k^3

is a multiple of

9

);

n

is divisible by

36

(

36k

is a multiple of

36

);

m \cdot n = 324k^4 = 108(3k^4)

, which is divisible by

108

. Counterexamples show

m

need not be a square (

m=72

) and

n

need not be divisible by

54

(

n=36

Verifies which properties hold universally versus which can fail.

Anahtar Kavram

Prime Factor Exponents in Perfect Powers and Divisibility Rules

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru