Let k be a positive integer with the prime factorization k = 2^x \cdot 3^y \cdot 5^z, wher… — GMAT

Let $k$ be a positive integer with the prime factorization $k = 2^x \cdot 3^y \cdot 5^z$ , where $x$ , $y$ , and $z$ are positive integers. If $k$ is divisible by both $18$ and $75$ , and $k$ has exactly $36$ positive integer divisors, what is the maximum possible value of $x + y + z$ ?

A
7
8Cevap
C
9
D
11
E
12

Cevap

The maximum possible value of

x + y + z

is 8.

The correct answer is 8. The prime factorization of

k

requires

x \ge 1

y \ge 2

, and

z \ge 2

because

k

is a multiple of

18 = 2^1 \cdot 3^2

and

75 = 3^1 \cdot 5^2

. The number of positive divisors is given by

(x+1)(y+1)(z+1) = 36

. Under the constraints

x+1 \ge 2

y+1 \ge 3

, and

z+1 \ge 3

, the factorizations of 36 into three factors yield the sums

x+y+z = 8

(from factors

2, 3, 6

) and

x+y+z = 7

(from factors

4, 3, 3

). Therefore, the maximum possible value is 8.

Adım Adım Çözüm

Determine the lower bounds for the exponents

x

y

, and

z

based on divisibility conditions.

Since

k

is divisible by

18 = 2^1 \cdot 3^2

, we must have

x \ge 1

and

y \ge 2

. Since

k

is divisible by

75 = 3^1 \cdot 5^2

, we must have

y \ge 2

and

z \ge 2

For a prime factorization to be divisible by another number, each prime factor's exponent in

k

must be at least as large as its corresponding exponent in the divisor.

Set up the equation for the total number of positive integer divisors of

k

(x + 1)(y + 1)(z + 1) = 36

, with constraints

x + 1 \ge 2

y + 1 \ge 3

, and

z + 1 \ge 3

The total number of positive divisors of

2^x \cdot 3^y \cdot 5^z

is given by

(x+1)(y+1)(z+1)

Find all valid integer factor triples

(A, B, C) = (x+1, y+1, z+1)

multiplying to 36 under the given constraints.

The valid triples

(A, B, C)

with

A \ge 2, B \ge 3, C \ge 3

are

(2, 3, 6)

(2, 6, 3)

, and

(4, 3, 3)

Testing factorizations of

36

2 \cdot 3 \cdot 6 = 36

2 \cdot 6 \cdot 3 = 36

, and

4 \cdot 3 \cdot 3 = 36

all meet the inequality bounds for each term.

Calculate the sum

x + y + z = (A + B + C) - 3

for each valid triple and identify the maximum.

For

(2, 3, 6)

x+y+z = (2-1) + (3-1) + (6-1) = 1 + 2 + 5 = 8

.
For

(2, 6, 3)

x+y+z = (2-1) + (6-1) + (3-1) = 1 + 5 + 2 = 8

.
For

(4, 3, 3)

x+y+z = (4-1) + (3-1) + (3-1) = 3 + 2 + 2 = 7

.
The maximum possible value is

8

Comparing all valid scenarios yields

8

as the maximum sum of exponents.

Anahtar Kavram

Calculating total positive integer divisors from prime factorizations and analyzing exponent constraints derived from divisibility conditions.

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