Question: 353 squared

Comment on 353 squared

DIABOLICAL!

Great solution! I used a different approach. Doing 2^2 - 1^2, 3^2 - 2^2, 4^2 - 3^2 I saw it is a sequence starting with 3 and adding 2 each number I increase. So, 3 + 2 * 351 = 705
gmat-admin's picture

Beautiful - very clever approach!!

Brent,

I used a different approach:
Let 352 = x
so, 353 = x+1

Doing so:
(x+1)^2-x^2
(x^2+2x+1)-x^2=2x+1
Substituting x = 352
2(352)+1=705

So I got ans. E

What do you think about?

Cheers,
Pedro
gmat-admin's picture

That's a great approach - well done, Pedro!

Unbelievable

I used a slightly different approach:

Instead of using difference of squares directly, I recognized 353^2 is the same as (352+1)^2

So your expression works out to be: (352+1)^2 - 352^2
Since we know that 352^2 will cancel out, you only solve the "2xy" (2*352*1) part of the equation and add the 1^2 to it, which gives us 705.
gmat-admin's picture

Very nice reasoning!

Hi Brent, this might sound absolutely crazy but I tried to simplify it. Taking 3^2 - 2^2 which is 9 - 4 = 5. I realize that 5 is just the sum of 3 + 2. the two numbers. Then I just 353 + 352 to get the correct answer. Does it make any sense or is it just witchcraft?
gmat-admin's picture

Pure witchcraft - burn the witch!!! :-)

I LOVE your approach!
Using smaller numbers, you saw a pattern and applied that pattern to the larger numbers in the question.

The main reason your approach worked was that the numbers were consecutive.
For example, we can't use the exact same approach if the numbers were two apart (e.g., 126² - 124²). HOWEVER, your approach of testing smaller numbers will still get you on the right track. Give it a try.

Cheers,
Brent

Is it incorrect to say (353)^2 - (352)^2 = (352 + 1)^2 - 352^2
= 352^2 + 1^2 - 352^2
= 1^2
= 1
gmat-admin's picture

That would certainly make things easier, except (352 + 1)² does not equal 352² + 1².
In general, (x + y)² ≠ x² + y²

We can also test a few cases to convince ourselves.
For example, is it true that (5 + 1)² = 5² + 1²?
Simplify both sides: 6² = 25 + 1
Simplify again: 36 = 26
No good. So, (5 + 1)² ≠ 5² + 1²

Likewise, is it true that (7 + 3)² = 7² + 3²?
Simplify both sides: 10² = 49 + 9
Simplify again: 100 = 58
No good. So, (7 + 3)² ≠ 7² + 3²

Let's take a closer look at what (x + y)² SHOULD equal.
(x + y)² = (x + y)(x + y)
= x² + xy + xy + y²
= x² + 2xy + y²

Likewise, (352 + 1)² = (352 + 1)(352 + 1)
= 352² + 2(352)(1) + 1²

Cheers,
Brent

Thanks!
gmat-admin's picture

Hey Brent, one question:

https://gmatclub.com/forum/which-of-the-following-expressions-can-be-wri...

Besides checking back with the answer choices, could we know whether II is an integer?
Cause possibly, 10 times 0,1 is also an integer, so is 4 times 0,25.
So just by checking on whether one nr is not an integer no necessarily gives us certainty, right?

gmat-admin's picture

Hmmm, that's odd.
I made a small edit to your question, and it now looks like I asked the question.

That's correct.
Just because one of the two values is an integer (and the other is not an integer), we can't make any conclusions on whether their product is an integer.

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