Playing with Numbers Test

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Playing with Numbers Test - 22

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following pairs are co-prime?

A
$$348 \ and \ 296$$

B
$$56\ and \ 97$$

C
$$3025 \ and \ 4920$$

D
None

Solution

If the H.C.F of two numbers are $$1$$ they are said to be co-prime
Factors of $$56$$ = $$1,2, 4, 8, 28, 56$$
Factors of $$97$$ = $$1, 97$$
Common factor = $$1$$
Since, both have only one common factor, i.e. $$1$$
therefore, they are co-prime numbers.
Question 2
1 / -0

Example for co-prime numbers is

A
$$27,14$$

B
$$18,16$$

C
$$9,18$$

D
$$11,77$$

Solution

The two numbers which have only $$1$$ as their common factor are called co-primes.

Out of the given pair of numbers,
Factors of $$ 27 $$ are $$ 1,3,9,27$$

Factors of $$ 14 $$ are $$ 1,2,7,14 $$
Their common factor is $$ 1 $$.
Hence $$ 25,14 $$ are co-prime numbers.
Question 3
1 / -0

What is the HCF of $$24, 36$$ and $$92$$ ?

A
$$24$$

B
$$36$$

C
$$12$$

D
$$4$$

Solution

Factors of $$ 24 = 1, 2, 3, 4, 6, 8, 12, 24 $$
Factors of $$ 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 $$
Factors of $$ 92 = 1, 2, 4, 23, 46, 92 $$
Common factors are $$ = 1, 2, 4 $$
$$\therefore $$ HCF $$ = 4 $$
Question 4
1 / -0

How many numbers from 1 to 100 are there each of which is not only exactly divisible by 4 but also has 4 as a digit ?

A
7

B
10

C
20

D
12

Solution

The numbers from $$1$$ to $$100$$ which are exactly divisible by $$4$$ are:-
$$4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100$$

But each number should have $$4$$ as its digit.

Therefore, the required numbers are $$4, 24, 40, 44, 48, 64, 84$$.
Clearly, there are 7 such numbers.

Hence, there are 7 numbers from $$1$$ to $$100$$ each of which is not only exactly divisible by $$4$$ but also has $$4$$ as a digit.
Question 5
1 / -0

A number $$\displaystyle a_{1}a_{2}a_{3}a_{4}a_{5}$$ is divisible by 9 if
(i) $$\displaystyle a_{1}+a_{2}+a_{3}+a_{4}+a_{5}$$ is divisible by $$9.$$
(ii) $$\displaystyle a_{1}-a_{2}+a_{3}-a_{4}+a_{5}$$ is divisible by $$9.$$

Which of the above statements is/are correct?

A
only (i)

B
only (ii)

C
both (i) and (ii)

D
neither (i) nor (ii)

Solution

The given number is divisible by $$9$$, if sum of
its digits is divisible by $$9$$.
$$\therefore \displaystyle a_{1}+a_{2}+a_{3}+a_{4}+a_{5}$$ is divisible by 9

So, option A is correct.
Question 6
1 / -0

Example for co-prime numbers is

A
$$25, 14$$

B
$$18, 16$$

C
$$9, 18$$

D
$$11, 77$$

Solution

Two numbers are co-prime if their highest common factor (HCF) is $$1$$. For example, the numbers $$2$$ and $$3$$ are co-primes to each other.

(a) Let us find the HCF of $$25$$ and $$14$$ by factorizing these numbers:

$$25=5\times 5\\ 14=2\times 7$$

So, HCF$$(25,14)=1$$ and therefore, $$25$$ and $$14$$ are co-prime numbers.

(b) Let us find the HCF of $$18$$ and $$16$$ by factorizing these numbers:

$$18=2\times 3\times 3\\ 16=2\times 2\times 2\times 2$$

So, HCF$$(18,16)=2$$ and therefore, $$18$$ and $$16$$ are not co-primes.

(c) Let us find the HCF of $$9$$ and $$18$$ by factorizing these numbers:

$$18=2\times 3\times 3\\ 9=3\times 3$$

So, HCF$$(9,18)=3\times 3=9$$ and therefore, $$9$$ and $$18$$ are not co-primes.

(d) Let us find the HCF of $$11$$ and $$77$$ by factorizing these numbers:

$$11=11\times 1\\ 77=11\times 7$$

So, HCF$$(11,77)=11$$ and therefore, $$11$$ and $$77$$ are not co-primes.

Hence, $$25$$ and $$14$$ are co-prime numbers.
Question 7
1 / -0

Which number is divisible by $$6$$ ?

A
$$213$$

B
$$468$$

C
$$621$$

D
None of these

Solution

$${\textbf{Step - 1: Write divisibility test for 6}}{\textbf{.}}$$
$${\text{The whole number has to be divisible by }}2.{\text{ A number is divisible by }}2{\text{ if the unit place}}$$
$${\text{digit of the number is even, i}}{\text{.e}}{\text{. }}0,2,4,6,{\text{ and }}8.$$
$${\text{The whole number has to be divisible by 3}}{\text{. A number is divisible by 3 if the sum}}$$
$${\text{of all digits of the number is a multiple of 3 or the sum is exactly divisible by 3}}{\text{.}}$$
$${\textbf{Step - 2: Check the example using the divisibility test}}{\textbf{.}}$$
$${\text{Option A, as unit digit is not even so not divisible by 2}}{\text{.}}$$
$${\text{Now sum up all the digits 2 + 1 + 3 = 6}}.$$
$${\text{As 6 is multiple of 3, but it does not pass divisibility test of 2 so}}$$$${\text{it is}}$$
$${\text{not divisible by 6}}.$$
$${\text{Option B, as unit digit is divisible by 2 also when summed up 4 + 6 + 8 = 18 is}}$$
$${\text{divisible by 3}}{\text{.}}$$
$${\text{Hence, 468 is divisible by 6 as it fulfills both the condition}}{\text{.}}$$
$${\text{Option C, as unit digit is not divisible by 2}}{\text{. When summed up 6 + 2 + 1 = 9}}{\text{.}}$$
$${\text{It is divisible by 3 but not divisible by 2 hence not divisible by 6}}{\text{.}}$$
$${\textbf{Hence, the number (B) 468 is divisible by 6}}{\textbf{.}}$$
Question 8
1 / -0

Which of the following number is divisible by 8?

A
23, 624

B
39, 126

C
1,20,458

D
None of these

Solution

To check if a number is divisible by $$ 8 $$, we check if the last three digits of the number is divisible by $$ 8 $$ .

The last three digits of $$ 23,624 $$ are $$ 624 $$
Now, $$ 624 = 8 \times 78 $$

Hence as $$ 624 $$ is divisible by $$ 8 $$, the number $$ 23,624 $$ is divisible by $$ 8 $$.
Question 9
1 / -0

Fill in the blanks.
Every composite number can be expressed as a product of ____, and this factorization is unique except for the order in which the prime factors occur.

A
Co-primes

B
Twin primes

C
Primes

D
All of these

Solution
Question 10
1 / -0

Find the number of factors of $$512$$.

A
$$8$$

B
$$10$$

C
$$4$$

D
$$14$$

Solution

Factors of $$512=1,2,4,8,16,32,64,128,256,512$$
Therefore, number of factors of $$512=10$$.

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Re-Attempt Weekly Quiz Competition

Playing with Numbers Test - 22

Result

Playing with Numbers Test - 22

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SHARING IS CARING

Question 1

$$348 \ and \ 296$$

$$56\ and \ 97$$

$$3025 \ and \ 4920$$

None

Solution

Question 2

$$27,14$$

$$18,16$$

$$9,18$$

$$11,77$$

Solution

Question 3

$$24$$

$$36$$

$$12$$

$$4$$

Solution

Question 4

7

10

20

12

Solution

Question 5

only (i)

only (ii)

both (i) and (ii)

neither (i) nor (ii)

Solution

Question 6

$$25, 14$$

$$18, 16$$

$$9, 18$$

$$11, 77$$

Solution

Question 7

$$213$$

$$468$$

$$621$$

None of these

Solution

Question 8

23, 624

39, 126

1,20,458

None of these

Solution

Question 9

Co-primes

Twin primes

Primes

All of these

Solution

Question 10

$$8$$

$$10$$

$$4$$

$$14$$

Solution

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