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Odd numbers, square of integers...

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by Brent@GMATPrepNow » Mon Aug 10, 2015 11:35 am
yass20015 wrote:How many odd numbers between 10 and 1000 are the square of integers ?
The answer is 14. I found only 12... any ideas/approach? thanks
Start with: what's the largest integer that's the square of an integer.
Well, 30² = 900
31² = 961
32² = 1024

Okay, so 961 (aka 31² is the largest ODD integer that is the square of an integer.
This means 29² will also meet this condition.
Also, 27² meets the condition.
As does 25²
etc.

The smallest ODD number that is the square of an integer is 25 (5²)

So, how many ODD integers are in the list from 5 to 31?
We have 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 and 31

TOTAL = 14
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by nikhilgmat31 » Wed Aug 12, 2015 3:58 am
once we found 5 & 31 as min & max numbers

we can use Airthmatic progression

L = a + (n-1)d

31 = 5 + (n-1)2

26/2 = n-1

n = 13 + 1 = 14
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by [email protected] » Wed Aug 12, 2015 9:06 am
Hi yass20015,

You'll see some questions on the GMAT that are "just about the math." However, you might not need to do lots of math to come up with the correct answer.

Here, we're asked for the number of ODD numbers, between 10 and 1000 that are the squares of integers. From the answers, we know that there are at least 12, but no more than 16 numbers that fit this description. Since we're given a range to work with, we should try to find the smallest and largest values, then we can just count up the number in between.

Starting at the lower end of the range, we know that....
3^2 = 9
5^2 = 25

So the smallest number is 5

Now, the upper end of the range will take a bit more work to figure out...
30^2 = 900 (we're not allowed to use even numbers though, so this calculation was just to help us "zero in" on the upper limit.
31^2 = 961 (you'll need to be comfortable doing this math by hand)

Notice how 31^2 is 61 'more' than 30^2. Since it's so close to 1,000 there's really no reason co calculate 32^2 or 33^2 (as they'll both be even 'further away').

With a lower limit of 5 and an upper limit of 31, we just need to total up the ODD numbers in this range:
5 7 9
11 13 15 17 19
21 23 25 27 29
31

Total = 14

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