For how many n in {1, 2, 3, ..., 100} is the tens digit of n 2 odd? (A

Hi Noshad,

This is not a typical GMAT question, so there is a very small chance that you can see such a question on test day. But it is a good question that plays on pattern recognition that the GMAT tests on. The big takeaway from the question here, is that even though the question looks daunting, starting small and forming patterns will get you an answer.

The fastest way to solve the question is to know some properties of perfect squares (the GMAT will never expect a test taker to know these properties! Remember the GMAT is a test of reasoning, not math!)

1. If unit digit of a number is 2, 3, 7 and 8 then the number cannot be a perfect square.

2. If unit digit of a perfect square is 1 then tens digit has to be Even
E.g. 81, 121, 441, 961 all are Perfect Square having unit digit 1 and tens digit is even.

3. If unit digit of a perfect square is 4 then tens digit has to be Even.
E.g. 64, 144, 484 all are Perfect Square having unit digit 4 and tens digit is even.

4. If unit digit of a perfect square is 5 then tens digit has to be 2
E.g. 25, 225, 625,1225 all are Perfect Square having unit digit 5 and tens digit is 2.

5. If unit digit of a perfect square is 0 then tens digit has to be 0.
E.g. 100,400,900 all are Perfect Square having unit digit 0 and tens digit is 0.

6. If unit digit of a perfect square is 9 then ten’s digit has to be Even.
E.g. 49, 169, 529 all are Perfect Square having unit digit 9 and tens digit is even.

This leaves us with the last case of having a perfect square whose unit digit is 6.

7. If unit digit of a perfect square is 6 then tens digit has to be Odd. e.g. 256, 576, 676, 1296 etc.


The question tests us on this property. In the squares from 1 to 10 the only 2 numbers that gives us a 6 as the units digit are 4^2 and 6^2. So it will suffice for us to find out how many numbers from 1 to 100 will end with a 4 or a 6. This will be a total of 20.

In a very unlikely scenario that this question is tested on the GMAT, the best way to approach this question is to start small and look for patterns.

Knowing the values of all the squares from 1 to 20 and the cubes from 1 to 10 is something that we encourage test takers to keep in mind as they will help reduce time in calculation. In this question if we know the squares of 1 to 10 we should be able to figure out a pattern and extrapolate this pattern to a 100 values.

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100.

Here we see that the only 2 numbers that give us an odd tens digit are 4 and 6. So this is 2 numbers in the first set of 10, so in the next set of 10 we will have another 2 i.e. 14^2 and 16^2. This gives us an answer of 2 * 10 = 20

Hope this helps!

Aditya
CrackVerbal Academic Team