Direct Divisibility MCQ Quiz - Objective Question with Answer for Direct Divisibility - Download Free PDF

Given:

9-digit number: 5x79856y6

This number is divisible by 36.

x and y are natural numbers.

Formula Used:

A number is divisible by 36 if it is divisible by both 4 and 9.

1. For divisibility by 4, the last two digits must form a number divisible by 4.

2. For divisibility by 9, the sum of all digits must be divisible by 9.

Calculation:

Last two digits are 6y6, so for divisibility by 4:

y6 must be divisible by 4.

Possible values for y: 1, 3, 5, 7, 9 (natural numbers)

Sum of digits: 5 + x + 7 + 9 + 8 + 5 + 6 + y + 6 = 46 + x + y

For divisibility by 9, (46 + x + y) must be divisible by 9.

Considering y = 9 (largest possible value):

46 + x + 9 = 55 + x

55 + x must be divisible by 9.

Possible values for x = 8 (as 55 + 8 = 63)

so, x = 8 and y = 9

Now,

Neagtive value of  √(2x + y) =  √(2 × 8 + 9) =  √25 = - 5

The correct answer is option 4.

Given:

Numbers: 1475, 3471, 5418, 4795

Formula Used:

A number is divisible by 9 if the sum of its digits is divisible by 9.

Calculation:

For 1475:

Sum of digits = 1 + 4 + 7 + 5 = 17

17 is not divisible by 9.

For 3471:

Sum of digits = 3 + 4 + 7 + 1 = 15

15 is not divisible by 9.

For 5418:

Sum of digits = 5 + 4 + 1 + 8 = 18

18 is divisible by 9.

For 4795:

Sum of digits = 4 + 7 + 9 + 5 = 25

25 is not divisible by 9.

Correct Option: Option 3

Solution Statement: The number 5418 is divisible by 9 as the sum of its digits (18) is divisible by 9.

Given:

Number when divided by 6519 gives remainder 97

So, Number = 6519 × q + 97 (for some integer q)

Formula used:

If N ≡ r (mod a), then N mod b = r mod b (when b divides a or b < a)

Calculation:

Required: Remainder when same number is divided by 53

So, we check: 97 ÷ 53

⇒ 97 = 53 × 1 + 44

∴ The remainder is 44.

Given:

Number = 6745K2

We need to find the value of K such that 6745K2 is divisible by 9

Formula used:

A number is divisible by 9 if the sum of its digits is divisible by 9

Calculation:

Sum of digits = 6 + 7 + 4 + 5 + K + 2 = 24 + K

We need 24 + K divisible by 9

Try K = 3

⇒ 24 + 3 = 27 (divisible by 9)

∴ The correct answer is option (3): 3

Given:

A 4-digit number 5k21 is divisible by 9

Formula used:

A number is divisible by 9 if the sum of its digits is divisible by 9

Calculation:

Sum of digits = 5 + k + 2 + 1

⇒ Sum of digits = 8 + k

For 8 + k to be divisible by 9:

⇒ 8 + k = 9

⇒ k = 1

∴ The correct answer is option (2).

Concept used:

If the last 2 digits of any number divisible by 4, then the number is divisible by 4

Calculation:

According to the question, the numbers are

2332, 2552, 4664, 2772, 6776, 4884, 2992, and 6996

So, there are 8 such numbers in the form abba, divisible by 4

∴ The correct answer is 8

Mistake Points

If you are considering an example ending with 20,

then, 'abba' will be '0220', and 0220 is not a four-digit number. 

Similarly in the case of the example ending with 40,60,80.

Calculation:

625 + 626 + 627 + 628

Taking 6²⁵ commons from the expression:

⇒ 625(6⁰ + 6¹ + 6² + 6³)

⇒ 6²⁵(1 + 6 + 36 + 216)

⇒ 625 × 259

After simplifying we get that the given expression is the multiple of 259.

Thus, the given expression is divisible by 259.

∴ The correct answer is option (4).

Given:

2727 + 27

Concept used:

Aⁿ + Bⁿ is divisible by (A + B) when n is odd.

Calculation:

Now, (2727 + 27)

⇒ (2727 + 1²⁷ + 27 - 1)

⇒ (2727 + 127) + 26

Here, according to the concept, (2727 + 127) is divisible by (27 + 1) i.e. 28.

Hence, the remainder = 26

∴ The remainder when 2727 + 27 is divided by 28 is 26.

Given:

A six-digit number Is divisible by 33

Formula used:

Dividend = divisor × quotient + remainder

Calculation:

Dividend = divisor × quotient + remainder

⇒ 33 × q + 0 = 33q

If 54 is added to the dividend then,

New number = 33q + 54

⇒ 3 × (11q + 18)

So, we can clearly say that the new number is divisible by 3.

∴ The correct option is 1.
Mistake Points 

Please note that this is the official paper of SSC and SSC has given the 3 as the correct answer, but 111111 is also the 6 digit number and if we add 54 it will be divisible by both 3 and 5.

Given:

The 8-digit number 123456xy is divisible by 8

Concept used:

If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Calculation:

So, 6xy should be divisible by 8

Now,

Possible numbers are 600, 608, 616, 624, 632, 640, 648, 656, 664, 672, 680, 688, 696

So, total of 13 possible pairs can be made

∴ The required answer is 13.

Calculation:

⇒ 412 + 413 + 414 + 415

⇒ 4¹² (1 + 4 + 4² + 4³)

⇒ 412 (1 + 4 + 16 + 64)

⇒ 412 × 85

⇒ 412 × 5 × 17

So the number is a multiple of 17.

∴ The correct answer is 17.

Calculation:

When we divided 87501 with 765 we get 291 as the remainder

So, the nearest number greater than 87501 = 87501 - 291 + 765

⇒ 87975

∴ The required answer is 87975.

Given:

The largest five-digit number = 99999

Concept used:

Dividend = (divisor × quotient) + remainder

Calculation:

The largest five-digit number = 99999

If we divide it by 88 then

Dividend = (divisor × quotient) + remainder

99999 = (88 × 1136) + 31

So the largest number which is divisible by 88 = (99999 - 31) = 99968

∴ The correct answer is 99968.

Given:

The nine-digit number 3422213AB is divisible by 99

Concept used:

Divisibility Rule of 9:

if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.

Divisibility Rule of 11:

If the difference of the sum of alternative digits of a number is divisible by 11, then the number itself is divisible by 11.

Calculation:

First, we check the divisibility with 9

⇒ 3 + 4 + 2 + 2 + 2 + 1 + 3 + A + B = 17 + A + B

The possible values of (A + B) are:

A + B = 1 ------- (1)

and A + B = 10 ------- (2)

Now, we check the divisibility with 11

⇒ (3 + 2 + 2 + 3 + B) - (4 + 2 + 1 + A) = 0

⇒ (10 + B) - (7 + A) = 0

⇒ A - B = 3 -------- (3)

And,

⇒ (3 + 2 + 2 + 3 + B) - (4 + 2 + 1 + A) = 11

⇒ (10 + B) - (7 + A) = 11

⇒ B - A = 8 ------------- (4) 

If we take equation (1) and (3) then;

A = (3 + 1)/2 = 2

B = (1 - 3)/2 = - 1

B = - 1 is not possible.

If we take equation (2) and (3) then;

A = (10 + 3)/2 = 13/2

Which is not possible. 

If we take equation (1) and (4) then;

B = (8 + 1)/2 = 9/2

Which is not possible.

If we take equation (2) and (4) then;

B = (10 + 8)/2 = 9

A = (10 - 8)/2 = 1

Putting the required value in the given equation:

⇒ 2A + B = 2 × 1 + 9 = 11

∴ The correct answer is 11.

Given:

350 + 926 + 2718 + 928 + 929 

Calculation:

350 + 926 + 2718 + 928 + 929

⇒ 3⁵⁰ + 3⁵² + 3⁵⁴ + 3⁵⁶ + 3⁵⁸

⇒ 3⁵⁰(1 + 3² + 3⁴ + 3⁶ + 3⁸)

⇒ 350(1 + 9 + 81 + 729 + 6561)

⇒ 350 × 7381

⇒ 350 × (11² × 61)

Hence, the given number is divisible by 11.

∴ 350 + 926 + 2718 + 928 + 929 is divisible by 11.