If x and y are positive integers such that 3x + 5y = 101 and x is a prime number, what is … — GRE General Test

If $x$ and $y$ are positive integers such that $3x + 5y = 101$ and $x$ is a prime number, what is the maximum possible value of the product $xy$ ?

A
38
B
108
C
112
170Answer
E
180

Answer

170

The equation

3x + 5y = 101

requires

101 - 3x

to be a multiple of 5. For positive integers

x

and

y

x

must be less than 34 and leave a remainder of 2 when divided by 5 (meaning

x

ends in 2 or 7). Testing the primes meeting this condition yields three valid pairs:

(2, 19)

with product 38,

(7, 16)

with product 112, and

(17, 10)

with product 170. Thus, 170 is the maximum product.

Step-by-Step Solution

Isolate

5y

and analyze modular divisibility by 5.

5y = 101 - 3x

. For

y

to be an integer,

101 - 3x

must be divisible by 5. Since

101 \equiv 1 \pmod 5

, we require

3x \equiv 1 \pmod 5

, which simplifies to

x \equiv 2 \pmod 5

Properties of modular arithmetic determine which values of

x

yield integer values for

y

Determine the boundary constraints and prime condition for

x

Since

y

is a positive integer (

y \ge 1

3x < 101 \implies x \le 33

. The prime numbers

x \le 33

that satisfy

x \equiv 2 \pmod 5

(or end in 2 or 7) are

x = 2

x = 7

, and

x = 17

Only primes satisfying both the inequality bound and divisibility criteria need to be tested.

Calculate the corresponding values of

y

and products

xy

for each prime candidate.

For

x = 2

5y = 101 - 6 = 95 \implies y = 19

, product

xy = 2 \times 19 = 38

.
For

x = 7

5y = 101 - 21 = 80 \implies y = 16

, product

xy = 7 \times 16 = 112

.
For

x = 17

5y = 101 - 51 = 50 \implies y = 10

, product

xy = 17 \times 10 = 170

Evaluating all valid candidate pairs allows identifying the absolute maximum product.

Identify the maximum product.

The maximum possible value of

xy

is 170.

Comparing 38, 112, and 170 confirms that 170 is the greatest value.

Key Concept

Linear Diophantine equations with prime integer constraints

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