Divisibility

Question 1

When can we say that a number is divisible by other?

Answer 1

When you get an integer out of the division.
18 is divisible by 3 because 18 : 3 = 6, an integer
12 is not divisible by 8 because 12 : 8 is not equal to an integer

Question 2

When is a number divisible by 2?

Answer 2

When it is an even integer (i.e., all integers that end in 0, 2, 4, 6, or 8)

Question 3

When is a number divisible by 3?

Answer 3

When the sum of integer’s digit is divisible by 3
E.g.: 147
1 + 4 + 7 = 12. 12 is divisible by 3, so 147 is divisible by 3

Question 4

When is a number divisible by 5?

Answer 4

When the integer ends in 0 or 5

Question 5

When is a number divisible by 9?

Answer 5

When the sum of the integer’s digit is divisible by 9
E.g.: 288
2 + 8 + 8 = 18. 18 is divisible by 9, so 288 is divisible by 9

Question 6

When is a number divisible by 10?

Answer 6

When the integer ends in 0

Question 7

What is a factor of a number?

Answer 7

Is a number which is a divisor of a number, that divide the number evenly
E.g., 2 is a divisor of 6
2 goes into 6 evenly
2 is a factor of 6
6 is divisible by 2
6 is a multiple of 2

Question 8

What are the factor pairs of 105?

Answer 8

1 - 105
3 - 35
5 - 21
7 - 15

Question 9

What is a prime number?

Answer 9

Is a number that can only be divided by 1 and itself

Question 10

The concept of prime applies only to positive integers. True or false?

Answer 10

True.

Question 11

Is 1 a prime number?

Answer 11

No, because it has only one factor (itself), not two factors

Question 12

What are the even prime numbers?

Answer 12

Only 2. Every even number greater than 2 has at least one more factor besides 1 and itself, namely the number 2

Question 13

Which are the smaller primes?

Answer 13

2, 3, 5, 7, 11, 13, 17 and 19

Question 14

What is a prime factor tree?

Answer 14

The factor tree represents only the prime factors of a number (it doesn’t show all factors of a number)
105 =
5 * 21
5 * 3 * 7

Question 15

Is every number divisible by the factors of its factors?

Answer 15

Yes. 12 is divisible by 6, and 6 is divisible by 3. Therefore, 12 is divisible by 3 as well as by 6.
You can use prime factors as building blocks to find another factors, e.g.:
The prime factors of 150 are 2, 5, 3, and 5
2 x 3 = 6, and 5 x 5 = 25, so 6 and 25 are also factors of 150.
3 x 5 x 5 = 75, so 2 and 75 are also factors of 150.
2 x 3 x 5 = 30, so 5 and 30 are also factors of 150
2 x 5 x 5 = 50, so 3 and 50 are also factors of 150

Question 16

How to build factors out of primes?

Answer 16

Every factor of a number in the factor pair can be expressed as the product of some or all of the number’s prime factors. E.g.:
60
1 - 60 (2x2x3x5)
2 - 30 (2x3x5)
3 - 20 (2x2x5)
4 (2x2) - 15 (3x5)
5 - 12 (2x2x3)
6 (2x3) - 10 (2x5)

Question 17

For each problem, the following is true: x is divisible by 24. Determine whether each statement below “must be true”, “could be true”, or “cannot be true”.
1) x is divisible by 6
2) x is divisible by 9
3) x is divisible by 8

Answer 17

x =
24 * ? =
2 * 12 * ? =
2 * 2 * 6 * ? =
2 * 2 * 2 * 3 * ?
1) Must be true. 6 is a factor of 24 (2x3)
2) Could be true. 9 has two 3 (3x3) as prime factors, so as long x have another 3 as prime factor, this could be true
3) Must be true. 8 has 2x2x2 as prime factors and x also has three 2 as prime factors

Question 18

If one number has all the prime factors that other given number, it must be true that the first number is divisible by the second.

Answer 18

True. E.g.:
x is divisible by 3 and by 10. So x is divisible by 15
x = 3 * ?
x = 10 * ? = 2 * 5 * ?
x = 3 * 2 * 5 * ? (if you know two factors of x that have no primes in common, combine te two trees into one)
15 = 3 * 5
So x has all the prime factors that 15 has, then x is divisible by 15

Question 19

How to find the least common multiple (LCM)?

Answer 19

When the two trees do not share any common prime factors, you can keep all of their prime factors in the combined tree. If you multiply those prime factors together, you’ll find the least common multiple, or LCM. E.g.:
LCM of 3 and 10.
3 = 3 * 1
10 = 2 * 5
LCM = 3 * 2 * 5
LCM = 30
The least common multiple of two numbers is defined as the least number that is a multiple of both A and B. So when you know that two numbers, say A and B, don’t share any common prime factors, then the LCM of A and B is equal to A x B

Question 20

Why is the LCM important?

Answer 20

Because if x is divisible by A and by B, then x is divisible by the LCM of A and B.
Please note that when the factors don’t share any prime factors, their LCM is always equal to their product (e.g.: 3 and 10 LCM is 30).
However, when two numbers do share prime factors, their LCM will always be less than their product because you have to strip out the overlap prime factors between them (e.g.: 6 and 9 share prime factors, so their LCM is not 6 * 9 = 54. In fact, their LCM (18) is less than 54).

Question 21

What to do when you have factors of x with primes in common?

Answer 21

When two starting numbers share a common factor, you can count the common factor only once, not twice. So you can combine to the least common multiple

Question 22

If x is divisible by 8, 12, and 45, what is the greatest number that x must be divisible by?

Answer 22

x = 8 * ? = 2 * 2 * 2 * ?
x = 12 * ? = 3* 2 * 2 * ?
x = 45 * ? = 3 * 3 * 5 * ?
x = 2 * 2 * 2 * 3 * 3 * 5 * ?
LCM = 72 * 5 = 360