What is the greatest common divisor (GCD) of 2^3 \times 3^2 \times 5 and 2^2 \times 3^3? — GMAT

What is the greatest common divisor (GCD) of $2^3 \times 3^2 \times 5$ and $2^2 \times 3^3$ ?

$36$ Cevap
B
$72$
C
$180$
D
$1080$
E
$18$

Cevap

The greatest common divisor (GCD) of

2^3 \times 3^2 \times 5

and

2^2 \times 3^3

36

The greatest common divisor (GCD) is calculated by taking the lowest power of each prime factor common to both expressions. For

2^3 \times 3^2 \times 5

and

2^2 \times 3^3

, the common prime factors are

2

and

3

. Taking the smallest exponents gives

2^2 \times 3^2 = 4 \times 9 = 36

Adım Adım Çözüm

Identify the common prime factors in both numbers.

The prime factors present in both numbers are

2

and

3

. The factor

5

is only present in the first number.

GCD requires prime factors that are shared by all target numbers.

Select the minimum exponent for each common prime factor.

For

2

\min(3, 2) = 2

. For

3

\min(2, 3) = 2

. For

5

\min(1, 0) = 0

The GCD contains only the prime powers that divide both numbers completely.

Multiply the resulting prime power factors to evaluate the GCD.

\text{GCD} = 2^2 \times 3^2 = 4 \times 9 = 36

Evaluating the prime factor powers gives the final integer value.

Anahtar Kavram

To find the Greatest Common Divisor (GCD) of numbers written in prime factorized form, take the product of each common prime factor raised to its smallest exponent across all the factorizations.

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