Divisibility

Prime Number

- Only divisible by 1 and itself - 1 is not prime - 2 is smallest and only even prime - Every number is either prime or not prime

Seven cards in a pile are numbered 1 through 7. One card is drawn. The units digit of the sum of the numbers on the remaining cards is 7. What is the number on the drawn card? (a) 1 (b) 3 (c) 5 (d) 6 (e) 7

1 2 3 4 5 6 7 What is the sum of all of these? 28 what minus 28 gives you a seven digit in the ones? 1! A

If n is divisible by both 196 and 15, is 270 a factor of n?

196 = 2, 2, 7, 11 15 = 3, 5 n = 2, 2, 7, 11, 3, 5, ? 270 = 3, 3, 3, 5, 2 For 270 to be a factor of n, needs all prime factors of 270, divides into 270 with no remainder Could be, not sure

Rules of divisibility

2 if even 3 if the sum of the digits is a multiple of 3 4 if the sum of the last 2 digits is a num div by 4 5 if the number ends in 5 or 0 (if not, not) 6 if ther number is divisble by 2&3 9 if the sum of the digits is divisble by a multiple of 9 10 if the number ends in 0

If N is a positive integer, then what is the least value of N for which N! Is divisible by 1000? 1 4 9 15 30

Divisible by 1000 means that N! Has all prime factors of 1000. Must have those building blocks. The answer is 15. You work backwards to solve the problem

If 6 and 14 are factors of y - must y be divisible by 21?

Factor - means that 6 and 14 divide into y with no remainder 6 = 2, 3 14 = 7, 2 y = ?, 2, 3, 7 ^ when you combine in factor tree, strip out common factors. you cant be sure how many except the max number it occurs. Yes

MULTIPLE

List of numbers that are products of another number - Multiply a given number by another - Infinite Multiplication - Greater than number

On the last day of a one week sale, customers numbered 149 through through 201 were waited on. How many customers were waited on that day? a 51 b 52 c 53 d 153 e 152

On medium and difficult GMAT problems, it is always not the easiest answer like this 201-149 = 52 WIBNI Make up numbers where you see the principle in place 9 > 21 21-9 = 12 but actually 13! Bc this question is inclusive - actually one more than the difference - as opposed to exclusive and using the word "between" So the answer is 52+1 = 53 C

Application of GCF

Split 2 things w diff values into equal parts Simplifying fractions and polynomials 2 more things into thier greatest groupings Arrange into rows/groups "Dividing into greatest equal segments"

The sum of positive integers x and y is 17 - if x only has 2 factors and y is a multiple of of 5 - which of the following is a possible value of x? 3, 7, 12

x is a prime, less than 17 x = 1, ? y = 5, ? 7 is the only prime - x must be 7 y = 10

Rule 1

If one variable contains all the same factors as another variable then x is a multiple of y

GCF

Largest number that is a factor of both numbers Product of all the prime factors that appear in the prime factorization of both numbers

The sum of the positive odd integers less than 50 is subtracted from the sum of the positive even integers less than or equal to 50. What is the resulting difference? (a) 0 (b) 25 (c) 50 (d)100 (e) 200

Make up numbers! 1 2 3 7 9 = 25 2 4 6 8 = 20 5*4 = 25 2 4 6 8 10 50 1 3 5 7 9 49 1 1 1 1. 1. 1 = 25

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T? (a) None(b) One(c) Seven(d) Thirteen(e) More than thirteen

Multiples are infinite! Start here: What is in these sets? s = 7, 7*2, 7*3 t = 13, 13*2, 13*3 Where would these overlap? 7*13 = 91 7*2*13*2 and on and on! E

LCM Application

SMALLEST, NO REMAINDER OF 2 are the keys to remember here - An event repeating over an over again - Purchase or order - get multiple types of items and have enough - When something will happen at the same time - When something will line up - Soonest 2 events will occur at the same time

How to find all the factors of a number?

Use a factor pair table - this gives you all the numbers that divide into another with no remainder. This is a finite list, unlike multiples. 1) Always start with one and the number 2) Use divisor rules to run down 2-10 3) Go through until the numbers on each side of the column run into each other

If n is the product of 2, 3 and 2 digit prime #, how many of its factors are greater than 6.

WORK BACKWARDS - plug in a number Use 11 Make a factor pair table Answer is 4

The key to solving divisibility problems...

Working down to the prime factors

4, 21, 55 are factors of n. Is n divisible by 154?

Yes

If a is not a multiple fo 30 but ab is, what is the smallest possible value of ab?

a = ? ab = ?, 3, 5, 2 a or b could have 3. 5, 2 the smallest of these is 2 so the smallest a could be is 2

Divisor

- A number divided by another number (divisor) - 9 / 3 = 3 is the divisor

FACTOR

- Divides Evenly - Leaves no remainder behind when divided - Every number has at minimum of 2 factors (1 and itself)

LCM

- The smallest number that is a multiple of 2 or more numbers - The smallest number that is divisible by of 2 or more numbers - Divisible by means that LCM will be divided by both original number and leave no remainder

If q is divisible by 2, 6, 9, 12, 15, 30 - is q divisible by 8?

- We need 1 factor with 2 8's to be sure - None of these have it - Could be but not sure

14 and 3 divide evenly into n Is 12 a factor of n?

14 = 7, 2 3 = 3 n = ?, 3, 7, 2 12 = 3, 2, 2 Could be, missing a 2 Must contain 2 2's in one factor

M is a multiple of 13 - is mn a muliple of 195 N has every factor that 45 has M is divisible by 18

195 = 5 3 13 N = ?, 3, 3, 5 (we also have m = 13 ?) M = ? 13, 9, 2 Just 1 is enough info to give you every factor of 195

Primes up to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

If n is a multiple of 35 and 44 is 14 a divisor of n?

35 = 7, 5 44 = 2, 2, 11 n = 7, 5, 2, 2, 11, ? Yes

The greatest common factor of 16 and the positive integer n is 4 and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210? 3,14,30,42, 70

42

GCF of 24 and 150

6

Dividend

A number that is to be divided by another number. 9/3 - 9 is the dividend

Rule 2 12 vs 30

But not always - must contain ALL THE SAME PRIME FACTORS. Can be in different numbers, but if one has a 5 and one does not then they are not multiples or factors.