# 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!

### 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.

### Hi,

Hi,

I did something quite different but would like your opinion. I took 345 and 355 as you taught us how to square number quickly that way. I used this technique and found 7000 difference for a difference of 10 then I divided this by 10 to have 700. There was only one answer with a 700 base, so I took 705. I know I didn't find the exact answer but is this approach valid or just luck?

### Unfortunately that approach

Unfortunately that approach was just good luck.

The question asks us to evaluate 353² - 352², but those values (353 and 352) don't appear anywhere in your solution.
So, it seems like you chose 345 and 355 because they're kind of close to 353 and 352.

If the question asked us to evaluate 354² - 353², would you still use 345 and 355 in your approach?

### OK, SO for this video I used

OK, SO for this video I used a different approach but I had to admit that your one is more simple.

so I let 353 = (352+1)^2 and get 252^2+1^2+704 then I get 704+1 which is 705