Number Properties

For unit digit problems, once you find cycle, divide exponent by # in cycle, the Remainder will be the the position of order

https://gmatclub.com/forum/units-digits-exponents-remainders-problems-175004.html

Divisibility of 6

(it passes both the 2 rule and 3 rule above) Divisible by 2 and 3? 114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes

How to find whether X is a multiple of 90? x/24=int0 thru x/27=int

1) Find LCM of 24 thru 27, inclusive 2) 24=2^3x3 25= 5^2 26= 2x 13 27=3^3 LCM: 2^3 x 3^3 x 5^2x13 2) X is a multiple of 90 Ex: 16 is a multiple of 4 4 is a factor of 16 X is a multiple of 90 ------ X will be bigger than 90 LCM of 90- 3^2x 2x5 LCM- (2^3 x 3^3 x 5^2x13)/ (2x 3^2x5)= is this divisible? 2^2x3x5x13=====so yes it is divisible. You will get a whole number.

0 is

neither prime nor composite

1 is

neither primer nor composite

2 is the

smallest prime number and only even prime number

15 is a multiple of 5 =

15 is divisible by 5— 15 can be divided evenly by 5

LCM- Go w/ opposite of name

take the greatest power of each prime factor. The lowest number that can be multiply to get.

GCF- Go w/ opposite of name

take the lowest power of each prime factor. Greatest building block of two numbers.

Least Common Multiple Help (LCM)

2: 0,2,4,6 3:0,3,6 6 is LCM------remember not 0 Large numbers------ 1) Prime tree of both numbers 135---- 3x3x3x5 135= 3^3x5 147= 3x7x7 147= 3x7^2 2)*MOST IMPORTANT STEP: PICK GREATEST EXPONENT OF EACH PRIME FACTOR 3^3 x 5^1 x 7^2 = 6615

Dividing smaller number by bigger number?

65/320=? Put decimal after "65." +add decimal to top of div line, then put numbers to right of decimal.

Divisibility of 11

Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11. 1364 (+1−3+6−4 = 0) Yes 913 (+9−1+3 = 11) Yes 3729 (+3−7+2−9 = −11) Yes 987 (+9−8+7 = 8) No

Remainder Formula

x= yQ+R Q=quotient - answer to division problem R= Remainder- Left over. 6/17. 0r6

Remainder formula*******

Dividend= Divisor x Quotient+ Remainder N=DQ+R Also, N/Q= D+ R/Q Where R/Q equals any multiple. 2r/2q, 3r/3q, 4r/4q N=Dividend D=Divisor Q=Quotient R= Remainder

Divisibility of 7

Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7. (We can apply this rule to that answer again) Ex: 672 (Double 2 is 4, 67-4=63, and 63÷7=9) Yes

Even Coefficient (2x)

Even number 2x= Even #

Odd Coefficient (3x)

Even or Odd number 3x= even or odd number

To be a perfect square

Every prime factor exponent must be even.

How to find the total number of FACTORS?

For Pos integer K= P^m x P^n.......... Total # of Factors K=(m+1)(n+1)....... Ex If N is the product of 1-10, EXCLUSIVE, How many Factors does integer N have? N: 2x3x4x5x6x7x8x9 4: 2x2 6: 2x3 8: 2x2x2 9: 3x3 2^7 x 3^4x5x7 (7+1)(4+1)(1+1)(1+1)= 8x5x2x2= 160 factors.

For unit digit problems see

Help: https://magoosh.com/gmat/2012/gmat-quant-finding-the-units-digits-of-large-powers/ Hard Problems: https://magoosh.com/gmat/2013/gmat-quant-difficult-units-digits-questions/

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

How to solve quickly? 3^k is a factor of 30! Take highest factorial (30) and divide by factor w/o "k" 30/3=10 10/3=3 w/o decimal place 3/3= 1 10+3+1=14

LCM will say

Lowest possible number Example: LCM of 8 and 12 is 8: 0,8,16,24 12:0,12,24 LCM is 24

Range

Max value-Min Value (2,4,5,6,8) 8-2= 6 The difference between two numbers in a set cannot be greater than the range.

Mode

Most common digit in set. (1,1,2,3,3) Mode is 1 and 3---- has two modes. (1,2,3,4) does not have a mode. (0,0,0,6,7,8) Mode is 0.

odd consecutive integers

N, N+1, N+3

even consecutive integers

N,N+2, N+4

r is a multiple "r" in problem

R/Q = 3/25 therefore R could be 3,6,9,12 Q is a multiple of 25----- 25, 50, 75 Where R/Q equals any multiple. 2r/2q, 3r/3q, 4r/4q

Least common Multiple

The LCM of two or more numbers can also be found using prime factorization. In order to do this, factor all of the numbers involved. For each prime number which divides any of them, take the largest power with which it appears, and multiply the results together. For example, to find the LCM of 8, 12 and 15, write: 8: 2^3 12: 3x 2^2 15: 5x3 Pick largest exponents--- 2^3x 3x5= 120

Divisibility of 4

The last 2 digits are divisible by 4 Ex: 1312 is (12÷4=3) Yes 7019 is not (19÷4=4 3/4) No

Divisibility of 5

The last digit is 0 or 5 Ex: 175 Yes 108 No

Divisibility of 2 ----- Ex 562

The last digit is even

Divisibility of 8

The last three digits are divisible by 8 Ex: 109816 (816÷8=102) Yes

Divisibility of 12

The number is divisible by both 3 and 4 (it passes both the 3 rule and 4 rule above) 648 (By 3? 6+4+8=18 and 18÷3=6 Yes) (By 4? 48÷4=12 Yes) Both pass, so Yes 524 (By 3? 5+2+4=11, 11÷3= 3 2/3 No) (Don't need to check by 4) No

Divisibility of 3

The sum of the digits is divisible by 3 Ex: 381 (3+8+1=12, and 12÷3 = 4) Yes 217 (2+1+7=10, and 10÷3 = 3 1/3) No

Divisibility of 9

The sum of the digits is divisible by 9 Ex: 1629 (1+6+2+9=18, and again, 1+8=9) Yes 2013 (2+0+1+3=6) No

Odd Even Problems

Write out----- E+E=E E+O=O O+O=E If you forget, use small numbers.

X^n has the same prime factors as

X

Prime Numbers

a number which can only be divided by itself and 1. (Has to be Positive) 2,3,5,7,11,13,17,19

2 and 3 are the only

consecutive numbers that are prime. But 2,3,5,7,11,etc are all consecutive prime numbers.