Arithmetic
I. Arithmetic Basics
1. Integers
Definition: Integers are whole numbers, they don’t include any parts or any pieces.
There are categories of numbers.
- Counting/Natural Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9,…
- Whole Numbers: similar to counting/natural numbers, but includes the number 0.
- Integers: include whole numbers and the negative counterparts: …,-4, -3, -2, -1, 0, 1, 2, 3, 4,…
a. Number Line
The number line is a visual representation to help see the relationship between numbers. → The right side of the line is larger than the left side.
b. PEMDAS
PEMDAS is a way to remember the Order of Operations.
- Parenthesis (brackets)
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
c. Integers in Intervals
1. Number of Integers in Interval
To find the number of integers in an interval, simply use the formula: (Last Number − First Number) / (difference between two consecutive numbers) + 1
Example: To find the number of integers from 27 to 84, inclusive: (84 – 27) / 1 + 1 = 58
NOTE: “inclusive” means you include the 27 and 84 in the range and “exclusive” means you don't. If you see the word exclusive, you would instead do: (83 – 28) / 1 + 1 = 56
2. Sum of Integers in Interval
To get the sum of all integers from a to b:
- Find the number of integers in the interval (using the formula in part a)
- Calculate the sum of the first and last number.
- Multiply the numbers from steps 1 and 2 together and divide by 2.
Example: What is the sum of all integers from 30 to 101?
- (101 – 30) / 1 + 1 = 72
- 30 + 101 = 131
- 72 × 131 / 2 = 4716
d. Three Consecutive Integers
| SUM of three consecutive integers | PRODUCT of three consecutive integers |
|---|---|
| Sometimes even, sometimes odd | Always even |
| Always a multiple of 3 | Always a multiple of 3 and 6 |
| Sometimes a multiple of not only 4 but also 8 |
2. Divisibility Rules
- All integers are divisible by 1. (1 is a factor of every integer.)
- An integer is divisible by 2 if it is even.
- An integer is divisible by 3 if the sum of the digits in that integer is also divisible by 3.
- For example, 816 is divisible by 32 because 8+1+6=15, a number divisible by 3.
- An integer is divisible by 4 if the last two digits in that integer are also divisible by 4.
- For example, 56,234,332 is divisible by 4 because 32 is divisible by 4.
- Note that if the last two digits are 00, the entire number is divisible by 4 because 00 is divisible by 4.
- An integer is divisible by 5 if the last digit in the number is a 5 or a 0.
- An integer is divisible by 6 if it is also divisible by 2 and 3.
- Another way to think about this is all even numbers that are divisible by 3 are also divisible by 6.
- An integer is divisible by 8 if the last three digits of the number are divisible by 8.
-
- Note that if the number ends in 000, it is divisible by 8.
-
- An integer is divisible by 9 if the sum of the digits of the number is divisible by 9.
- For example, 5985 is divisible by 9 because the sum of the digits is 27, a multiple of 9.
- An integer is divisible by 10 if the last digit in the number is 0.
- An integer is divisible by 11 if the integer can satisfy the following condtions:
- For each of the number in the integer, assign a sign + then —, then +, alternately.
- If the total of the all the numbers in the integer equals 0 or equals a number that is divisible by 11, then that number is divisible by 11.
Prime Numbers
To determine whether the integer n is prime, take the square root of n and see if it is divisible by any prime numbers lower than that square root value.
3. Factors and Factorials
a. Factors
**Definition:* If an integer is divisible by a certain number, then that certain number is a factor of that integer. e.g. The integer 36 is divisible by 4, 9, -6,… → All of those numbers are factors of 36.
Note: Negative integers can also be factors, but the GRE usually asks only about positive factors.
An effective way to list out all the factors of an integer is using the Nifty Factor-Finding System.
Example:
b. Factorials
Example:- 4! = 4 × 3 × 2 × 1
- 10! = 10 × 9 × 8 x … x 2 × 1
1. Factors of Factorials
Factorials have a lot of factors!
Example: 8! has the obvious factors of 8, 7, 6, 5, 4, 3, 2, and 1. But it has many more:
- 56 is also a factor of 8! because 7 × 8 = 56
- 30 is also a factor of 8! because 6 × 5 = 30
- Even 1440 is a factor of 8! because 8 × 6 × 5 × 2 × 2 = 1440
1. Calculate # of n in Factorial - WHEN N IS A PRIME NUMBER $$ Formula: Factorial / n^x $$ Example: n = 3. How many 3s are there in 5174! Keep dividing the factorial by the multiple and note down the result (whole number). Do this until you get 0.
- 5174! / 3 = 1724
- 5174! / 3^2 = 574
- 5174! / 3^3 = 191
- 5174! / 3^4 = 63
- 5174! / 3^5 = 21
- 5174! / 3^5 = 7
- 5174! / 3^7 = 2
- 5174! / 3^8 = 0 → stop
→ The number of 3s in 5174! = 1724 + 574 + 191 + 63 + 21 + 7 + 2 = 2582.
**2. Calculate # of n in Factorial - WHEN N IS NOT A PRIME ** $$ Formula: Factorial / (a * b)^x ;;;;;;;;;;;; (where ; a * b = n ;, a ;and ; b; are; both ; prime) $$ Example: n = 15. How many 15s are there in 200! Because 15 is not a prime number, we rewrite the problem. 15 = 3 × 5.
- To make one 15, we need 3 and one 5.
- We’re going to find more powers of 3, which means 5 is the limiting factor. → The question becomes: How many 5s are there in 200!. Because 5 is a prime number, we can use formula from part 1.
- 200! / 5 = 40
- 200! / 5^2 = 8
- 200 / 5^3 = 1
- 200 / 5^4 = 0 → stop
→ The number of 15s in 200! = 40 + 8 + 1 = 49
2. NON Factors of Factorials
Example: Find the smallest integers that are not factors of 20!.
- Step 1: Find the prime numbers greater than 10. e.g. 23, 29, 21, 37, 31, etc. → All are NOT factors of 20!.
- Step 2: To find the non-prime non-factors, simply find the multiples of previously identified primes.
- 23: 46, 69, 92
- 29: 58, 87, 116
Note: This trick really only works for factorials greater than or equal to 10!. If it is a smaller factorial, it’s better to use a brute-force solution, list them all out.
c. Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) of two numbers is the largest factor common to both numbers.
For example, if we list out all the positive factors of 12 and 18, we will see that 6 is the GCF:
- 12: 1, 2, 3, 4, 6, 12
- 18: 1, 2, 3, 6, 9, 18
When the two numbers are not unreasonably large, we can simply list out all the positive factors of each to identify the GCF.
4. Multiples
If n is an integer, the numbers 0n, 1n, 2n, 3n, 4n and so on are multiples of n. Multiples can of course be negative.
Example: Multiples of 70 include 0, 70, 140, 210, 280 and so on.
a. Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two numbers is the smallest positive multiple they share in common. We have to specifically identify “positive” multiple here because of course 0 is a multiple of every integer.
For example, if we list out all the positive multiples of 1212 and 1818, we will see that 36 is the LCM:
- 12: 12, 24, 36, 48
- 1818: 18, 36, 54, 72
When the two numbers are not unreasonably large, we can simply list out all the positive multiples of each to identify the LCM.
b. Number of Multiples in Interval
1. From 1 to n
Divide n by the multiple in question and take the whole number result.
Example: How many multiples of 31 are there from 1 to 1,000?
1000 / 31 = ~ 32.26. → 31 multiples.
2. From n to m
Use the formula: (last multiple — first multiple) / multiple in question + 1
**Example: ** The number of multiples of 3 from 29 to 112 is:
(111 – 30) / 3 + 1 = 28
c. Sum of Multiples in an Interval
Example: Find the sum of all multiples of 3 from 1 to 500.
Step 1: Find the number of multiples of 3 from 1 to 500.
- 500 / 1 = 166 (multiples)
Step 2: List out the first three and the last three multiples in the interval.
- 3, 6, 9, …, 492, 495, 498. → Every “pair” in this list adds up to 501.
Step 3: There are 166 multiples (step 1), giving us 166 / 2 = 83 pairs. → The sum of all multiples of 3 from 1 to 500 is: 83 × 501 = 41, 583.