Problem Solving

lucas211 wrote:Hello everybody
Would appreciate a little help on the following question:

When a certain perfect square is increased by 148, the result is another perfect square. What is the value of the original perfect square?

a) 1296
b) 1369
c) 1681
d) 1764
e) 2500

lucas211 wrote:Hi Mitch

Thank you for your answer.
Would you be kind to elaborate on the following steps:

Thus:
x² - y² = -148

(x+y)(x-y) = (74)(-2)

(x+y)(x-y) = (36+38)(36-38)

I am having troubles with understanding how we get to find the (74)(-2) and afterwards the (36+38)(36-38).

Thanks in advance

GMATGuruNY wrote:

lucas211 wrote:Hi Mitch

Thank you for your answer.
Would you be kind to elaborate on the following steps:

Thus:
x² - y² = -148

(x+y)(x-y) = (74)(-2)

(x+y)(x-y) = (36+38)(36-38)

I am having troubles with understanding how we get to find the (74)(-2) and afterwards the (36+38)(36-38).

Thanks in advance

Since x² and y² are perfect squares, x and y are positive integers.

(x+y)(x-y) = (74)(-2).

This equation implies that x+y = 74 and x-y = -2.
In other words, x and y are two positive integers with a sum of 74 and difference of -2.
Some students will be able to deduce x=36 and y=38 by trial and error.

We could also solve for x and y algebraically.
Adding together x+y = 74 and x-y = -2, we get:
(x+y) + (x-y) = 74 + (-2)
2x = 72
x = 36.

Substituting x=36 into x+y = 74, we get:
36+y = 74
y = 38.

lucas211 wrote:

GMATGuruNY wrote:

lucas211 wrote:Hi Mitch

Thank you for your answer.
Would you be kind to elaborate on the following steps:

Thus:
x² - y² = -148

(x+y)(x-y) = (74)(-2)

(x+y)(x-y) = (36+38)(36-38)

I am having troubles with understanding how we get to find the (74)(-2) and afterwards the (36+38)(36-38).

Thanks in advance

Since x² and y² are perfect squares, x and y are positive integers.

(x+y)(x-y) = (74)(-2).

This equation implies that x+y = 74 and x-y = -2.
In other words, x and y are two positive integers with a sum of 74 and difference of -2.
Some students will be able to deduce x=36 and y=38 by trial and error.

We could also solve for x and y algebraically.
Adding together x+y = 74 and x-y = -2, we get:
(x+y) + (x-y) = 74 + (-2)
2x = 72
x = 36.

Substituting x=36 into x+y = 74, we get:
36+y = 74
y = 38.

Hi Mitch

And we know that (x+y) is 74 and (x-y) is -2 (and not for example 148 and -1 or 37 and -4 respectively) because:
the prime factors of 148 is 2,2,37, and
148, 1 and 37, 4 will not be able to give us the desired outcome with integers.
for example lets try with 148 and 1.

(x+y)(x-y) = (148)(-1)
(x+y)(x-y) => here we can´t make it fit while x and y have to be integers.

the same with 37 and -4

is this correct understood?

Thanks