Let f(n) represent the number of positive factors of a positive integer n. If n = 2^x \tim… — ACT

Let $f(n)$ represent the number of positive factors of a positive integer $n$ . If $n = 2^x \times 3^y$ , where $x$ and $y$ are positive integers such that $x + y = 7$ , what is the maximum possible value of $f(n)$ ?

Answer: 20

Answer

The maximum possible value of

f(n)

20

The number of positive factors of

n = 2^x \times 3^y

(x + 1)(y + 1)

. Since

x

and

y

are positive integers that sum to 7, the possible pairs for

(x, y)

are

(1, 6)

(2, 5)

, and

(3, 4)

(and their reversals). The products

(x+1)(y+1)

for these pairs are

2 \times 7 = 14

3 \times 6 = 18

, and

4 \times 5 = 20

. The maximum value is 20.

Step-by-Step Solution

Determine the formula for the number of factors of

n

f(n) = (x + 1)(y + 1)

For any positive integer expressed in its prime factorization

p^a \times q^b

, the total number of positive factors is

(a + 1)(b + 1)

Identify the possible values for

x

and

y

given the constraints.

(x, y) \in \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\}

The exponents

x

and

y

must be positive integers (

x, y \geq 1

) that sum to 7.

Calculate

(x + 1)(y + 1)

for each pair to find the maximum product.

The maximum product is

4 \times 5 = 20

Evaluating the products:

2 \times 7 = 14

3 \times 6 = 18

, and

4 \times 5 = 20

. The product of two integers with a fixed sum is maximized when the integers are as close as possible.

Key Concept

Calculating the number of positive factors of an integer from its prime factorization and maximizing the count under exponent constraints.

Estimated Time:1m 30s

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