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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Playing with Numbers Test - 22

Result

Playing with Numbers Test - 22

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SHARING IS CARING

Question 1

348 and 296348 \ and \ 296348 and 296

56 and 9756\ and \ 9756 and 97

3025 and 49203025 \ and \ 49203025 and 4920

None

Solution

Question 2

27,1427,1427,14

18,1618,1618,16

9,189,189,18

11,7711,7711,77

Solution

Question 3

242424

363636

121212

444

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

25,1425, 1425,14

18,1618, 1618,16

9,189, 189,18

11,7711, 7711,77

Solution

Question 7

213213213

468468468

621621621

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

888

101010

444

141414

Solution

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$348 \ and \ 296$

$56\ and \ 97$

$3025 \ and \ 4920$

$27,14$

$18,16$

$9,18$

$11,77$

$24$

$36$

$12$

$4$

$25, 14$

$18, 16$

$9, 18$

$11, 77$

$213$

$468$

$621$

$8$

$10$

$4$

$14$