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]]>**Prime numbers**

2 is the smallest prime number

2 is the only prime even number

1 is not a prime

3, 5, 7 are the only three consecutive odd integers that are each a prime.

**Digits**

In the integer 5,432, 5 is the thousands digit, 4 is the hundreds digit, 3 is the tens digit, and 2 is the units digit.

In a multiple-digit number, there are 10 possible units digits (0 – 9). In the first digit, i.e. the digit farthest to the left, there are only nine possible digits. A zero as the first digit would not be valid. For instance, 0,453 is not a four-digit number. It’s – if anything – the three-digit number 453.

**Divisibility Rules**

3: Add up the digits in any number. If they are multiple of 3, then the number is divisible by 3. For example, 258 is divisible by 3 because 2 + 5 + 8 is 15. And 15 is a multiple of 3.

4: If the tens and units digits of integer X is a multiple of 4, then integer X is divisible by 4. For example, 364 is divisible by 4, because 64 is a multiple of 4.

5: If a number ends in 5 or 0, it is divisible by 5.

6: If the rules to divisibility to 2 and 3 are fulfilled, then a number is divisible by 6. One hundred and eleven is not divisible by 6. Two hundred and forty six, because it is divisible by both 2 and 3, is divisible by 6.

9: If the sum of the digits of a number is divisible by 9, then that number is divisible by 9. For example, adding up the digits in 4,536, we get 4+5+3+6 = 18. Because 18 is divisible by 9, the number 4,536 is also divisible by 9.

**Odds (so to speak) and Ends**

Zero is an even integer

Integers can be negative numbers (-3, -6, -17, etc.)

0! = 1

x^0 = 1

The problems below can be solved by referencing the above. Good luck!

**Practice Problems**

1. What is the smallest integer that can be divided by the product of a prime number and 7 while yielding a prime number?

(A) 7

(B) 14

(C) 24

(D) 28

(E) 35

2. The sum of five consecutive even integers is 20. What is the product of the median of the series and the smallest integer in the series?

(A) 12

(B) 10

(C) 6

(D) 4

(E) 0

3. Which of the following integers is NOT divisible by 3?

(A) 624

(B) 711

(C) 819

(D) 901

(E) 969

Answers:

1. D

2. E

3. D

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]]>If you’ve ever come across a number with an exclamation mark in front of it, that’s the factorial sign, e.g. 5!.

Factorials don’t come up too often on the GRE, and when they do, it’s usually on harder problems. Still, you don’t want to be faced with an exclamation sign next to a number and then exclaim yourself, “What in the world!” (For the more expletive prone you can use your imagination!)

But no need to worry—just think of the factorial sign as a countdown. Whichever number is next to the factorial just count down to 1, multiplying each number together. So if you see 5!, all this means is 5 x 4 x 3 x 2 x 1 = 120.

Factorials are sometimes seen explicitly (as in the problem below.) Oftentimes, you will need to use them when solving a difficult subset of math problems known as combination/permutations.

For now, try this problem:

Column A | Column B |
---|---|

7! | 6! + 6! + 6! + 6! + 6! + 6! + 6! |

- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given

You can work this out the long way: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1. Then you can do the same for column B, multiplying out and then adding the 6!s. Of course, if you catch yourself doing too much math always know there is a shorter way. In fact, quantitative comparison is testing your logical approach, so do as little math as possible.

Here, we can see that column B is made up seven 6!, or 7(6!), which equals 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7!. Therefore the two columns are equal.

Next time I touch on factorials will be in the context of permutation/combination problem, but as long as you understand the above you will be able to approach these more difficult problems.

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