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MGMAT Divisibility & Primes
MGMAT GMAT Divisibility & Primes
61
Mathematics
Graduate
10/07/2010

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Cards

Term
What is an integer?
Definition

A whole number without a fractional part.

Can be POSITIVE, NEGATIVE or the number 0.

Term
What is the quotient?
Definition
The result of dividing two integers
Term
When is an integer DIVISIBLE by another number?
Definition
When the integer can be divided by the number with an INTEGER result i.e. NO REMAINDER
Term

 

8 / 2 = 4...

 

a) What can we say about 2 in this instance?

b) What can we say about 8 in this instance?

Definition

 

a) 2 is a DIVISOR or FACTOR of 8

b) 8 is a MULTIPLE of 2 and a MULTIPLE OF 4. 

8 is DIVISIBLE by 2 and by 4.

Term

 

An integer is divisible by 2 if...

Definition

 

It is EVEN.

 

If the ones digit is 0,2,4,6 or 8.

Term

 

An integer is divisible by 3 if...

Definition

 

The SUM of the integer's DIGITS is DIVISIBLE by 3

Term

 

An integer is divisible by 4 if...

Definition

 

The integer is divisible by 2 TWICE

 

or

 

the LAST TWO digits are divisible by 4 (useful for larger numbers)

Term

 

An integer is divisible by 5 if...

Definition
The integer ENDS in 0 or 5
Term

 

An integer is divisible by 6 if...

Definition

 

It is divisible by BOTH 2 and 3.

 

e.g. 48 is divisible by 2 (ends with an 8, which is EVEN), and 3 (4+8 = 12, which is divisible by 3).

Term

 

An integer is divisible by 8 if...

Definition

 

It is divisible by 2 THREE TIMES, or the LAST THREE DIGITS are divisible by 8 (for larger numbers)

Term

 

An integer is divisible by 9 if...

Definition

 

The SUM of the integer's DIGITS is divisible by 9

Term

 

An integer is divisible by 10 if...

Definition

 

It ends in 0

Term

 

An integer has a ONES digit of 0. What is it divisible by?

Definition
10
Term

 

The sum of an integer's digits is 21. What is it divisible by? What is it NOT divisible by?

Definition

a) 3

b) 9 (2+1=3 is NOT divisible by 9)

Term

 

An integer x ends in 0 or 5. What is it divisible by? 

Definition
5
Term

 

 

An integer is divisible by 2 AND 3. What is it divisible by?

Definition
6
Term

 

The last 2 digits of a very large integer are divisible by 4. What is the integer divisible by?

Definition
4
Term

 

An integer divided by 2, two times, yields another integer. What must it be divisible by?

Definition

 

4

Term

 

The sum of an integer's digits is a multiple of 3. What must it be divisible by?

Definition
3
Term

 

1,234,567 is odd because?

Definition
7 is odd...
Term
2,345,678 is EVEN because?
Definition
8, being the last digit, is even
Term

 

An integer, divided by 10, yields a quotient ending in 5.

 

What is the quotient divisible by?

 

If the quotient was a 2 digit integer that ended in 0, what would it be divisible by?

Definition

 

a) 5

 

b) 10 or 5

Term

 

What is a FACTOR?

Definition

 

A positive integer that divides EVENLY into an integer with no remainder.

Term

 

What is a multiple of an integer ?

Definition

 

A multiple of an integer is formed by multiplying that integer by another integer.

Term

 

0 is a multiple of every number. 

 

TRUE or FALSE?

Definition

 

TRUE. 

 

 

Term

 

Why is 0 technically a multiple of any integer ?

Definition

 

Zero is technically a multiple of any number because that number times zero (an integer) equals zero (an integer).

Term

 

An integer is both a ___ and a  ___ of itself ?

 

How many integers is 1 a factor of?

Definition

FACTOR, MULTIPLE.

 

All integers.

 

Term

 

To find all the factors of a small number, you can use FACTOR PAIRS. 

 

Describe factor pairs for any integer, and describe this process.

Definition

 

Factor pairs for any integer are the pairs of factors that, when multipled together, yield that integer.

 

1. Start with the automatic factors (1 and the integer itself).

2. "Walk upwards" from 1, testing to see whether different numbers are factors of 72.

3. Keep walking upwards until all factors are exhausted.

 

 

Term

 

What is the mnemonic... Fewer ____, more _____ ?

Definition
Factors, Multiples.
Term

A factor is greater than the integer it is a factor of.

 

TRUE or FALSE ?

Definition

FALSE. 

 

Factors are ALWAYS less than or equal to the integer they are a factor of.

Term

 

 

A multiple is greater than the integer it is a multiple of.

 

TRUE or FALSE ?

 

Definition

 

 

TRUE. 

 

Multiples are ALWAYS greater than or equal to the integer they are a multiple of.

 

Term

 

12 = 3n, where n is an integer.

 

Is 12 a multiple of 3 and n?

 

Is 12 divisible by n?

 

Are 3 and n factors of 12?

 

How many times does n divide into 12?

 

Definition

 

YES

 

YES

 

YES

 

4

 

Term

12 items can be shared amongst 3 people so that each person has the same number of items... 

 

This is the same as saying:

Definition

 

12 is divisible by 3.

There are 4 items for each person.

12 is a multiple of the number of people.

12 is a multiple of the number of items in each group.

 

etc.

Term

 

(5 x n) + (3 x n) = 8 x 7...

 

What is n?

 

 

 

Definition

 

7

 

... because (5 + 3) x n = 8 x n = 8 x 7

 

 

 

Term

 

Adding/Subtracting multiples of N results in?

Definition

 

A multiple of N.

Term

 

If N is a divisor of x and y, then N is also a divisor of?

Definition

 

x + y

Term

 

What is a prime number?

Definition

 

A number larger than 1 with exactly 2 factors.

i.e. a number with NO factors other than 1 and itself.

Term

 

Why is 1 not a prime number?

Definition

 

Because it only has one factor: itself.

Term

 

The first 10 primes are?

Definition

 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Term

 


What is a good way to test if a number is a prime number?

Definition

 

CHECK THIS ONE IN O.G. and fill in asnwer

Term

 

Prime numbers are the building blocks of?

Definition

 

Integers

Term

 

What is the Factor Foundation Rule ?

Definition

 

 

If A is a factor of B, and B is a factor of C, then A is a factor of C

 

e.g. if 72 is div. by 12, then 72 is also div. by ALL ofthe factors of 12 (1,2,3,4 and 12).

 

or also you can say...

 

if 12 is a factor of 72, then all the factors of 12 are also factors of 72.

Term

 

You can build almost any factor of an integer out of the bottom level of the foundation built by its prime factors.

 

Why not all factors?

Definition

 

Because 1 is a factor of every number, but it is not prime.

 

DON'T forget to include 1 if asked for the number of unique factors of an integer, for example.

Term

 

 

Why are prime boxes useful?

Definition

 

It holds all the prime factors of a number (i.e. the lowest level building blocks).

 

You can use a prime box to test whether or not a specific number is a factor of another number.

 

e.g. is 27 a factor of 72? NO --> 72 = 2.2.2.3.3 ... 27 = 3.3.3 ... there are only two 3's in 72 so we can't make 27 from the prime factors of 72. Therefore 27 is NOT  factor of 72.

 

 

Term

 

 

 

The largest divisor of two or more integers is the?

Definition

 

 

GCF: Greatest Common Factor

Term

 

 

 

The smallest multiple of two or more integers is the?

Definition

 

 

 

LCM: Lowest Common Multiple

Term

 

 

 

When you reduce the fraction 9/12 to 3/4, you are doing what?

Definition

 

 

 

Dividing the numerator and denominator by 3, which is the GCF of both 9 and 12 (i.e. the largest divisor of 9 and 12)

 

 

Term

 

 

In a VENN Diagram of Overlapping Primes, 

the product of the primes in the shaded/overlapping area is the?

Definition

 

GCF

Term

 

 

 

In a VENN Diagram of Overlapping Primes, 

the product of ALL the unique primes in the diagram a is the?

 

Definition

 

 

 

 

LCM: Lowest Common Multiple

Term

 

 

 

When you add together 1/2 + 1/3 + 1/5, why do you convert the fractions to thirtieths?

Definition

 

 

 

 

Because 30 is the LCM of 2, 3 and 5

Term

 

In a VENN DTiagram of Overlapping Primes, the 2 outer regions should have primes in common

 

TRUE or FALSE?

Definition

 

FALSE

 

The 2 outer regions should have NO primes in common.

The common primes go in the shaded area of the VENN.

Term

 

 

 

If 2 numbers have NO primes in common, what is their GCF and LCM?

Definition

 

 

GCF = 1 (as 1 is the common factor of ALL positive integers)

 

LCM = the product of the 2 numbers

Term

 

 

17 / 5 = 3 Remainder 2

 

This is the same as saying...

Definition

 

 

17 = 15 + 2 = 3x5 + 2

 

17 is 2 more than 15

 

17 is 2 more than a multiple of 5

 

17 is 2 more than a multiple of 3

Term

 

 

 

x and y are positive integers and x = 10y + 5

 

why must x be divisible by 5 ?

Definition

 

 

Since both x and y are positive integers, 10y must be a multiple of 10. Also, since any multiple of 10 is also a multiple of 5, 10y is a multiple of 5. Therefore, x represents the sum of a multiple of 5 and 5. Since adding 5 to another multiple of 5 will always result in another multiple of 5, x must be a multiple of 5. As a result, xmust be divisible by 5 and will leave no remainder when divided by 5. The correct answer is A. 

Term

 

 

 

Explain a prime factor box, what it is for, and how to use it. Also, explain how to use a prime factor box for an unknown number and a known number.

Definition
Term

 

 

 

 

What can you infer from the fact that x has only 2 factors, and x >= 2?

Definition

 

 

 

 

x is prime. The factors MUST be 1 and x.

Term

 

 

 

 

What's an easy way to find the GCF and LCM of large numbers or of 3 or more numbers? Describe it.

Definition

 

Use the Prime Column technique.

 

1. Calculate the prime factors of each integer.

2. Create a column for each prime factor found within ANY of the integers.

3. Create a row for each integer.

4. In each CELL of the table, place the primer factor raised to a power. This power counts how many copies of the column's prime factor appear in the prime box of the row's integer.

 

Calculate GCF: Take the LOWEST count of each prime factor found across ALL the integers. i.e., count the SHARED PRIMES.

 

Calculate LCM: Take the HIGHEST count of each prime factor found across ALL the integers. This counts all the primes less the shared primes.

 

 

Term

 

 

4,620 is divisible by which of the following: 2,3,4,5,6,7,8,9, ?

 

13,131 is divisible by which of the following: 2,3,4,5,6,7,8,9, ?

Definition

4,620 is divisible by which of the following: 2,3,4,5,6,7,8,9, ?
- 2 - the number is even
- 3 - digits total 12, which is div. by 3
- 4 - the number formed by the lsat 2 digits, 20, is div. by 4
- 5 - the number ends in 0
- 6 - the number is divisible by 2 and 3 
- 7 - divide 4,620 by 7 and you get 660
- NOT 8 - the number formed by the last 3 digits, 620, is not divisible by 8
- NOT 9 - the total of the digits is 12, which is not divisible by 9



13,131 is divisible by which of the following: 2,3,4,5,6,7,8,9, ?
- 2 - the number is NOT even
- 3 - digits total 9, which is div. by 3
- NOT 4 - the number is not even
- NOT 5 - the number does not end in 0 or 5
- NOT 6 - the number is not even 
- NOT  7 - divide and you will see there is a remainder
- NOT 8 - the number is not even
-  9 - the digits total 9, which is divisible by 9

 

Term

// Divisibility and Addition/Subtraction //

 

... if you add a multiple of N to a non-multiple of N, the result is a (multiple of N / non-multiple of N?

 

... if you add two multiples of N , the result is a (multiple of N / non-multiple of N?


Definition

 

// Divisibility and Addition/Subtraction //

 

1. Non-multiple of N

 

 

2. Could be EITHER a multiple of N or a non-multiple of N

 

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