How many odd numbers between 10 and 1,000 are the squares of integers?

Hi All,

In these types of situations, you don't need to physically figure out every number in the set - you just have to figure out the smallest and the biggest, then you can 'count up' the number of terms in between (inclusive).

Here, we're asked for ODD numbers, between 10 and 1,000, that are SQUARES of INTEGERS. Since we're looking for ODD numbers, we're looking for the SQUARES of ODD numbers....

It helps to have certain perfect squares memorized and it helps to be able to do some basic multiplication by hand...

The first few perfect squares are....0, 1, 4, 9, 16, 25....

From this, we can determine the SMALLEST number that fits the above description: 25 (which is 5-squared). Now we have to figure out the BIGGEST number....

Here's where having strong arithmetic skills comes in handy (we can start with 'round' numbers to 'zero in' on the BIGGEST number)....

20^2 = 400 (which is way too small)
30^2 = 900 (which is getting close to 1,000)
31^2 = 961 (which is pretty close to 1,000)
33^2 = 1,089 (which is TOO BIG).

Now we know that 31-squared is the largest ODD number that fits the given description....Now we just have to count up the number of terms....

The ODD numbers are...
5-squared
7-squared
9-squared

11-squared
13-squared
15-squared
17-squared
19-squared

21-squared
23-squared
25-squared
27-squared
29-squared

31-squared

That's a total of 14 numbers.

Final Answer:
GMAT assassins aren't born, they're made,
Rich