A whole number without a fractional part.

Can be POSITIVE, NEGATIVE or the number 0.

8 / 2 = 4...

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

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

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.

An integer is divisible by 2 if...

It is EVEN.

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

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

The integer is divisible by 2 TWICE

or

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

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).

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

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

It ends in 0

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

a) 3

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

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

4

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

1,234,567 is odd because?

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?

a) 5

b) 10 or 5

What is a FACTOR?

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

What is a multiple of an integer ?

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

0 is a multiple of every number. 

TRUE or FALSE?

TRUE. 

Why is 0 technically a multiple of any integer ?

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

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

How many integers is 1 a factor of?

FACTOR, MULTIPLE.

All integers.

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

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

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.

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

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

TRUE or FALSE ?

FALSE. 

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

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

TRUE or FALSE ?

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?

YES

YES

YES

4

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

This is the same as saying:

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.

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

What is n?

7

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

Adding/Subtracting multiples of N results in?

A multiple of N.

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

x + y

What is a prime number?

A number larger than 1 with exactly 2 factors.

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

Why is 1 not a prime number?

Because it only has one factor: itself.

The first 10 primes are?

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

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

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

Prime numbers are the building blocks of?

Integers

What is the Factor Foundation Rule ?

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.

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?

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.

Why are prime boxes useful?

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.

The largest divisor of two or more integers is the?

GCF: Greatest Common Factor

LCM: Lowest Common Multiple

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

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)

In a VENN Diagram of Overlapping Primes, 

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

GCF

LCM: Lowest Common Multiple

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

Because 30 is the LCM of 2, 3 and 5

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

TRUE or FALSE?

FALSE

The 2 outer regions should have NO primes in common.

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

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

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

LCM = the product of the 2 numbers

17 / 5 = 3 Remainder 2

This is the same as saying...

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

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

why must x be divisible by 5 ?

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. 

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.

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

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

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

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.

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

// 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?