The product of the squares of two positive integers is 400. How many

Question

The product of the squares of two positive integers is 400. How many pairs of positive integers satisfy this condition?

A. 0
B. 1
C. 2
D. 3
E. 4

Kudos for a correct  solution .

VenoMfTw · Answer

IMO : D

Question Stem:   (a^2)(b^2)  = 400

(a^2)(b^2) = (2^4)(5^2)

a*b =  (2^2)(5^1)

a,b = (20,1) or (4,5) or (2,10)
Thus 3 pairs.

reto · Answer

First break down 200 into 20*20 and further into the prime factors 2*2*5*2*2*5. Now we are looking for all the possible pairs (2 numbers) of squares whose product results in 400.

1st: 2^2*10^2 (i.e. the first two 2's and two times 2*5 = 10)
2nd: 4^2*5^2 (i.e. two times 2*2 = 4 = 4^2 and 5^2).
3rd: 1^2*20^2 (i.e. two times 2*2*5 and 1^2 = 1)

Answer D.

GMATinsight · Answer

Given : (x^2)*(y^2) = 400

i.e. (x*y)^2 = (20)^2

i.e. (x*y) = (20)

1*20 = 20
2*10 = 20
4*5 = 20

i.e. 3 Possible pairs

Answer: option D

nitika231 · Answer

Ans: D - 3 pairs

(xˆ2)(yˆ2) = 400

xy = 20

20 = 1x20, 2x10, 4x5, 5x4, 10x2, 20x1

Cancel the repeats

This leaves us with exactly 3 options.

Hence, D

TimeTraveller · Answer

x^2 * y^2 = 400

(xy)^2 = 400

xy = 20

x  and  y  can take the following values:  (x,y) = (1,20), (2,10), (4,5) . Hence, Ans (D).

Bunuel · Answer

800score Official Solution:

400 = 2 × 2 × 2 × 2 × 5 × 5
Combine the prime factors in pairs.

400 = (2 × 2) × (2 × 2) × (5 × 5)
Now brake the factorization into two parts, each one will be a square.
The possible combinations are:
400 = (2 × 2) ×  
400 =   × (5 × 5)
But don't forget that 400 = 1 × 400, where 1 = 1². So we also have:
400 = (1 × 1) ×

Thus all the possible combinations of the factors that make the product of two squares are the following:
1² × 20² = 400
2² × 10² = 400
4² × 5² = 400

There are three possible pairs that fit the criterion.  The correct answer is D .

EddyEd · Answer

Can we reduce the expression to x*y=20 or x*y=5*2^2
And then approach it like this? The number of factors in 20 are 6, since we are looking for pair of factors, we divide de total number of factor by 2 and get 3 as a result.

Is this a right reasoning?

ScottTargetTestPrep · Answer

We can express 400 as a product of two numbers:

400 = 1 x 400 = 2 x 200 = 4 x 100 = 5 x 80 = 8 x 50 = 10 x 40 = 16 x 25 = 20 x 20

We see that of the products above, three products are products of two perfect squares:

1 x 400 = 1^2 x 20^2

4 x 100 = 2^2 x 10^2

16 x 25 = 4^2 x 5^2

Thus, there are 3 pairs of positive integers that satisfy the condition.

Alternate Solution:

If we denote the integers as n and m, we are given that (n^2)(m^2) = 400, which can be rewritten as (nm)^2 = 400 and simplified as nm = 20.

Thus, for any two integers whose product is 20, we get a pair of integers that satisfy the given condition. There are three such pairs: 20 x 1 = 10 x 2 = 5 x 4 = 20.

Answer: D

gadde22 · Answer

lets say 2 numbers are x and y.

So the equation is , x^2 * y^2 = 400 
 xy = 20

20 can be written as 1X20, 2X10, and 4X5, so 3 ways.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The product of the squares of two positive integers is 400. How many

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The product of the squares of two positive integers is 400. How many

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards