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Re: How many odd numbers between 10 and 1,000 [#permalink]

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11 Nov 2014, 22:27

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We know that only squares of odd numbers are odd 1^2=1, 3^2=9, 5^2=25........31^2=961,33^2=1089 we can see that squares of numbers between 5 and 31, inclusive, are the odd numbers lying between 10 and 1000 to find out how many numbers lie between 5 and 31, we may use sequences formula nth term=first term + (no of terms - 1)*common difference we have 31=5+(n-1)*2 Solving we get n=14 Hence Answer is choice (C)

So, that odd number could be any odd number from 5 to 31, inclusive: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, and 31.

14 numbers.

Answer: C.

Could you please advice on how to promptly figure out the upper and lower numbers for the set of odd number squares between 10 and 1000? Generally, how do you go about these types of questions?

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

A. 12 B. 13 C. 14 D. 15 E. 16

Total Perfect Squares from 1 through 1000 = 1^2 to 31^2 i.e. 31 Perfect Squares (with 15 even and 16 odd perfect squares)

Odd Perfect Squares from 1 through 1000 = 31 - 15 = 16 perfect Squares

Odd Perfect Squares from 1 through 10 = 1^2 and 3^2 = 2 perfect Squares

Total Odd Perfect Squares from 10 though 1000 = 16 - 2 = 14

Answer: option C
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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

So, that odd number could be any odd number from 5 to 31, inclusive: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, and 31.

14 numbers.

Answer: C.

Could you please advice on how to promptly figure out the upper and lower numbers for the set of odd number squares between 10 and 1000? Generally, how do you go about these types of questions?

You find the closest square you can think of and then go from there.

This is what I do when I want to find the first odd square after 10. I know that 9 is the square of 3. (an odd square) Next square of a number will be 4^2 which is 16 but this is an even square. So 5^2 = 25 will be the odd square closest to but greater than 10.

Similarly, when I want the odd square closest to but less than 1000, I will think about 900, the square of 30 (It is very easy to see squares of numbers ending in 0 or 5). It is an even square less than 1000. The next number 31 will have an odd square. On calculating, I find that 31^2 = 961. The square of 32 is 1024 and the square of 33 will be even higher.

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

A. 12 B. 13 C. 14 D. 15 E. 16

We have to find the squares at the extreme ends... 10----- square of 3 is 9.., so 4 is the lowest number in the series.. 1000.... clearly it is above 30^2=900.... check for 31, it is 961 and then it goes above 1000..

So we are looking for ODDs from 4 to 31... Total even and odd 31-4+1=28.. Odds =28/2=14 C
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Re: How many odd numbers between 10 and 1,000 [#permalink]

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16 Aug 2017, 12:27

Why don't we consider the negative integers here? For eg. 25 is the square of 5 and negative 5 also, so therefore in all there should be 14*2 that is 28 integers, right?

Why don't we consider the negative integers here? For eg. 25 is the square of 5 and negative 5 also, so therefore in all there should be 14*2 that is 28 integers, right?

That is true but you have to count the number of perfect squares, not the number of square roots. Whether 25 is the square of one or two numbers, it doesn't matter. It is a perfect square and counted only once.
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