# Question: 353 squared

## Comment on 353 squared

DIABOLICAL!

### Great solution! I used a

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 ### Beautiful - very clever

Beautiful - very clever approach!!

### Brent,

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

Cheers,
Pedro ### That's a great approach -

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

Unbelievable

### I used a slightly different

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. ### Very nice reasoning!

Very nice reasoning!

### Hi Brent, this might sound

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? ### Pure witchcraft - burn the

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

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 (352 +

Is it incorrect to say (353)^2 - (352)^2 = (352 + 1)^2 - 352^2
= 352^2 + 1^2 - 352^2
= 1^2
= 1 ### That would certainly make

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!

Thanks! ### Hey Brent, one question:

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? ### Hmmm, that's odd.

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.