# Free Playing with Numbers 02 Practice Test - 6th grade

Which of the following numbers is divisible by both 6 and 8?

A.

9638

B.

9640

C.

9642

D.

9648

#### SOLUTION

Solution : D

If a number is divisible by both 2 and 3, then it will be divisible by 6 also.

Divisibility by 2
Here all the numbers, 9638, 9640, 9642, 9648 are divisible by 2 as they end with one of these (0, 2, 4, 6, 8) numbers or simply as they are even numbers.

Divisibility by 3

9 + 6 + 3 + 8 = 26
9 + 6 + 4 + 0 = 19
9 + 6 + 4 + 2 = 21
9 + 6 + 4 + 8 = 27
Here only 21 and 27 are divisible by 3 hence only 9642 and 9648 are divisible by 3.
So, these two numbers are divisible by both 2 and 3 and hence divisible by 6.

Divisibility by 8
A number will be divisible by 8 only if the last three digits are divisible by 8.
The last 3 digits of only 9648, i.e. 648 is divisible by 8.

But last three digits of 9642, i.e. 642 is not divisible by 8.

9648 is divisible by both 6 and 8.

Two numbers having _____ as the only common factor are called ________ numbers.

A.

2, twin prime

B.

2, co-prime

C.

1, co-prime

D.

1, twin prime

#### SOLUTION

Solution : C

Two numbers having 1 as the only common factor are called co-prime numbers.

For example, 16 and 35 are co-prime number.
Factors of 16: 1, 2, 4, 8, 16
Factors of 35: 1, 5, 7, 35
Here, 1 is the only common factor.

What are the common factors of 75, 60 and 210?

A.

1, 3, 5 and 15

B.

1, 3, 5, 15 and 25

C.

1, 3, 15 and 25

D.

3, 6 and 15

#### SOLUTION

Solution : A

Factors of 75 are 1, 3, 5, 15, 25 and 75.

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60.

Factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

∴ Common factors of 75, 60 and 120 are 1, 3, 5 and  15.

Common multiples of 4, 5 and 11 are _________.

A.

220, 440 and 660

B.

220, 221 and 222

C.

210, 220 and 240

D.

200, 210 and 220

#### SOLUTION

Solution : A

Common multiples of any given set of numbers will always be the multiples of the LCM of the numbers.

Now, LCM of 4, 5 and 11 can be found as follows. Hence, LCM = 2×2×5×11=220
The multiples of 220 are 440, 660, 880...

Hence, in the given options, the common multiples of 4, 5 and 11 are 220, 440 and 660.

If two given numbers are divisible by a number, then their ____ is/are also divisible by that number.

A.

Product

B.

Difference

C.

Sum

D.

sum, difference and product

#### SOLUTION

Solution : D

If two given numbers are divisible by a number, then their sum, difference and product is also divisible by that number.

For example, the numbers 35 and 25 are both divisible by 5 and

their

i)difference (35-25 = 10) is also divisible by 5

ii)sum (35+25 = 60) is also divisible by 5.

iii)product(35×25=875) is also divisible by 5.

All numbers divisible by 4 must also be divisible by 8 and all numbers divisible by 8 must also be divisible by 4

A.

False

B. True

#### SOLUTION

Solution : A

If a number is divisible by a number 'n', it is also divisible by the factors of 'n'.

Factors of 8 = 1, 2, 4, 8

Any number divisible by 8 will also be divisible by 4.

But vice-versa is not true. For example 20 is divisible by 4 but not divisible by 8

Given that 312 is divisible by 26. It will also be divisible by ________.

A.

7

B.

9

C.

11

D.

13

#### SOLUTION

Solution : D

For any two numbers x and y, if x is divisible by y, then x is also divisible by each factor of y.

For example, 12 is divisible by 6, and 2 is a factor of 6. so, 12 is also divisible by 2.

Now, factors of 26 = 1, 2, 13, 26

312 is divisible by 26, hence it is also divisible by 13.

Statement 1: If two numbers are co-primes, at least one of them must be prime.

Statement 2: If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Statement 3: If a number exactly divides two numbers, then it must also divide the sum of both numbers.

A.

True, True, False

B.

False,False,True

C.

False, True, True

D.

True, True, True

#### SOLUTION

Solution : B

Take an example and visualize-

i)9 and 4 are co-prime but none of them are prime.

ii)Now, 14 divides 28, but it does not divide 12 and 16.

iii)4 divides 12 and 16 separately

4 also divides 12+16 = 28

What do you mean by prime factorization?

A.

Writing a number as a product of any other numbers.

B.

Writing a number as a product of only prime factors.

C.

Writing a number as a sum of only prime factors.

D.

Writing a number as a sum of any other numbers.

#### SOLUTION

Solution : B

Prime factorization is finding the factors of a number that are all prime.

Eg:- 12 = 2 × × 3

36 = 2 × 2 × × 3 etc.

Choose the correct option.

A.

72 = 2 × × × × × 3

B.

105 = 3 × × 7

C.

625 = 5 × × × × 5

D.

162 = 2 × × × × × 7

#### SOLUTION

Solution : B

72 = 2 × 2 × 2 × 3 × 3

105 = 3 × 5 × 7

625 = 5 × 5 × 5 × 5

162 = 2 × 3 × 3 × 3 × 3

Hence, only the factorisation of 105 is correct