Arithmetic & number Properties

Question 1

Multiplication & division of the same signs gives what result?

Answer 1

positive

Question 2

Multiplication & division of the opposite signs gives what result?

Answer 2

negative

Question 3

Is zero even?

Answer 3

Yes. It’s divisible by 2. 0 / 2 = 2 which is an integer.

Question 4

PEMDAS

Answer 4

parenthesis, then exponents, then with M or D (ensuring left to right) and then either A or S (left to right)

Question 5

Factoring process?

Answer 5

1. Group like terms (add or subtract to simplify)
2. Then pull out a common factor

Question 6

How do you know if an integer is even?

Answer 6

Even numbers are integers that end in 0,2,4,6 or 8

Question 7

How do you know if an integer is divisible by 3?

Answer 7

If the sum of the integers are divisible by 3

Question 8

How do you know if an integer is divisible by 5?

Answer 8

If the integer ends in 0 or 5

Question 9

How do you know if an integer is divisible by 9?

Answer 9

If the sum of the integers are divisible by 9

Question 10

What are prime numbers?

Answer 10

Integers that are only divisible by 1 and itself

Question 11

Integers are

Answer 11

Negative or positive whole numbers, including 0

Question 12

What are the primes between 0 and 50?

Answer 12

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

(tip is to remember to always exclude numbers that end in 0,2,4,6,8 or 5)

Question 13

If a is divisible by b and b is divisible by c, then a is also divisible by what?

Answer 13

a is divisible by c. For example 100 (a) is divisible by 20 (b). 20 (b) is divisible by 4 (c), so 100 (a) must be also divisible by 4 (c).

Question 14

If d has e and f as prime factors, then d is divisible by what in terms of e and f?

Answer 14

d is divisible by e,f and e x f

Question 15

If an unknown positive number (x) is divisible by 6. What do you know straight away about x?

Answer 15

If x is divisible by 6, it is also divisible by 3 and 2 (prime factors of 6)

Question 16

x is divisible by 6. Decide whether each statement must be true, could be true or cannot be true.

i. x is divisible by 3
ii. x is even
iii. x is divisible by 12

Answer 16

i. must be true (prime factors of 6 includes 3)
ii. must be true (prime factors of 6 is 2 which means must be even)
iii. could be true (draw separate prime tree for 12 and look at whether prime factors match). in this case there is one less 2 in x and 6 tree so we don’t know for sure.

Question 17

How do you approach questions that give you factors with no common primes?. For example *x is divisible by 3 and by 10. Decide whether the statments below must be true, could be true or cannot be true.

i. x is divisible by 2
ii. x is divisible by 15
iii. x is divisible by 45*

Answer 17

If there are no common overlapping prime factors:
* You can multiply those two factors to get the lowest common multiple (in this case, 3 x 10 = 30).
* combine their trees to form one tree and refer to image

i. must be true
ii. must be true
iii. could be true (missing a 3)

Two separate trees of x with prime factor 3 and x with prime factors 2 and 5 (stemming from the 10 divisor) share no common primes so therefore can be combined into one factor tree.
Lowest common multiple is 3 x 10 or 2 x 3 x 5

Then compare with the factor tree of 45.

Question 18

How do you approach questions that give you factors with primes in common?. For example *x is divisible by 6 and 9. Is it divisible by 54?

Answer 18

6 and 9 (when separate factor trees are drawn out) have a common prime of 3.
When factors share primes in common you need to find their lowest common multiply in this case, 18. Draw a tree for 18.

Question 19

When two numbers don’t share any prime factors, their LCM is always equal to

Answer 19

their product

For example: 3 and 10 don’t share any prime factors, so their LCM = 3 ×

Question 20

When two numbers do share prime factors, their LCM will always be

Answer 20

less than their product, because you have to strip out the overlap.

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.

Break the numbers into their primes and then take only the greater number of instances of any one particular prime. For example: 6 = 2 × 3 and 9 = 3 × 3
* How many 2’s should you take? The number 6 has one 2 and 9 has no 2’s, so take one 2.
* How many 3’s should you take? The number 6 has one 3 and 9 has two 3’s, so ignore the 6 and take the two 3’s from the 9.
The LCM is 2 × 3 × 3 = 18.

Question 21

If asked about two different variables (eg x and y), how do you approach finding their LCM and prime factors?

Answer 21

• Multiply them and then do a prime factor tree
• ie if x is divisible by 6 and y is divisible by 9, you do 6 x 9, 54, and do a tree based on that number

Question 22

How many more prime factors does the product of 42 × 120 have than the product of 21 × 24 ?

Answer 22

Question 23

Answer 23

Tip* to be divisible is the same as saying a multiple of

For integer a to be a multiple of 30, it would need to contain all of the prime factors of 30: 2, 3, and 5.
Since a is not divisible by 30 or not a multiple of 30, it must be missing at least one of these prime factors.
If ab is divisible aka a multiple of 30, b must supply any missing prime factors. The least possible missing prime is 2. If b = 2 and a = 15 (or any odd multiple of 15), then the initial constraints will be met: ab will be divisible by 30, but a by itself will not be.

Question 24

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

Answer 24

Maybe. To be divisible by 8, q needs three 2’s in its prime factorization. Because there might be some overlapping factors of 2, you cannot simply count all the numbers that contain 2 as a prime factor.

You can tell that q doesn’t necessarily have to be divisible by 8, since it is only guaranteed to have two 2’s, not the three 2’s required to create 8.)

Question 25

If p is a prime number, and q is a non-prime integer, what are the minimum and maximum numbers of factors p and q can have in common?

Answer 25

• Maximum will be 2 as prime integers are only divisible by 1 and themselves
• every integer has to be divisible by 1 as they are whole numbers!
• P and q can have two factors in common. If q is divisible by p ie p = 3 and q = 12, they share the prime factors of 1 and 3.
• minimum shared factors = 1
• maximum shared factors = 2

Question 26

If n is the product of 2, 3, and a two-digit prime number, how many of its factors are greater than 6 ?

Answer 26

If n is the product of 2, 3, and 11, then n = 66, and its factors are (1, 66), (2, 33), (3, 22), and (6, 11).

There are four factors greater than 6: 11, 22, 33, and 66.

The answer will always be four regardless of which two digit prime you choose.