# Number System & Divisibility Rules Questions

09 May 2023

In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals.

## Types of Numbers-

1. Natural Numbers - n > 0 where n is counting number; [1,2,3...]

2. Whole Numbers - n ≥ 0 where n is counting number; [0,1,2,3...]

3. 0 is the only whole number which is not a natural number. Every natural number is a whole number.

4. Integers - n ≥ 0 or n ≤ 0 where n is counting number;...,-3,-2,-1,0,1,2,3... are integers.

5. 0 is neither positive nor negative integer.

6. Even Numbers - n / 2 = 0 where n is counting number; [0,2,4,...]

7. Odd Numbers - n / 2 ≠ 0 where n is counting number; [1,3,5,...]

8. Prime Numbers - Numbers which is divisible by themselves only apart from 1.

9. Co-Primes Numbers - Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.

## Divisibility-

Following are tips to check divisibility of numbers.

1. Divisibility for 2 - A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.

2. Divisibility for 3 - A number is divisible by 3 if sum of its digits is completely divisible by 3.

3. Divisibility for 4 - A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.

4. Divisibility for 5 - A number is divisible by 5 if its unit digit is 0 or 5.

5. Divisibility for 6 - A number is divisible by 6 if the number is divisible by both 2 and 3.

6. Divisibility for 8 - A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.

7. Divisibility for 9 - A number is divisible by 9 if sum of its digits is completely divisible by 9.

8. Divisibility for 10 - A number is divisible by 10 if its unit digit is 0.

9. Divisibility for 11 - A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.

## Number System & Divisibility  Questions

A) 542622125

B) 584638125

C) 544638125

D) 584632125

A) 9

B) 4

C) 8

D) 2

A) 9944

B) 9900

C) 9988

D) 9999

A) 3210

B) 4200

C) 3025

D) 3060

### Q5) A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

A) 13

B) 59

C) 35

D) 37

#### Solution:

Let the original number be 'a'
Let the divisor be 'd'
Let the quotient of the division of a by d be 'x'
Therefore, we can write the relation as a/d=x and the remainder is 24.
i.e., a=dx+24
When twice the original number is divided by d,2a is divided by d.
We know that a=dx+24. Therefore, 2a=2dx+48
The problem states that  (2dx+48)/d leaves a remainder of 11.
2dx is perfectly divisible by d and will, therefore, not leave a remainder.
The remainder of 11 was obtained by dividing 48 by d.
When 48 is divided by 37, the remainder that one will obtain is 11.
Hence, the divisor is 37.

### Q6)  The product of 4 consecutive even numbers is always divisible by:

A) 600

B) 768

C)  864

D) 384

#### Solution :

To solve this question, we need to know two facts.
Fact 1:
The product of 4 consecutive numbers is always divisible by 4!.
Fact 2:
Since, we have 4 even numbers, we have an additional 2 available with each number.
Now, using both the facts, we can say that the product of 4 consecutive even numbers is always
divisible by,
=(24)×4!=(24)×4!
=16×24
=384

### Q7) What is the minimum number of square marbles required to tile a floor of length 5 metres 78 cm and width 3 metres 74 cm?

A) 176

B) 187

C) 540

D) 748

#### Solution:

The marbles used to tile the floor are square marbles.
Therefore, the length of the marble=width of the marble. the length of the marble=width of the marble.
As we have to use whole number of marbles, the side of the square should a factor of both 5 m 78
cm and 3m 74. And it should be the highest factor of 5 m 78 cm and 3m 74.
5 m 78 cm = 578 cm and 3 m 74 cm = 374 cm.
The HCF of 578 and 374 = 34.
Hence, the side of the square is 34.
The number of such square marbles required,
=(578×374)/(34×34)
=17×11= 187 marbles

### Q8) What number should be subtracted from x3+4x2−7x+12, if it is to be perfectly divisible by x+3?

A) 42

B) 39

C) 13

D) None of these

#### Solution:

According to remainder theorem when dfracf(x)x+a, then the remainder is f(−a).
In this case, as x+3 divides x3+4x2−7x+12–k  perfectly (k being the number to be subtracted), the
remainder is 0 when the value of x is substituted by -3.
i.e., (−3)3+4(−3)2−7(−3)+12−k=0
or −27+36+21+12=k
or k= 42

### Q9) Find the remainder when 289 is divided by 89?

A) 1

B) 2

C) 87

D) 88

#### Solution :

When we take successive powers of 2 and find their remainders, we get the following cyclic patterns of cycle length 11.
viz 2,4,8,16,32,64,39,78,67,45,1
i.e 211 leaves a remainder 1.
Thus, 289=(211)8(2) leaves a remainder of 2.

### Q10) What is the remainder when 37 is divided by 8?

A) 1

B) 2

C) 3

D) 5

#### Solution :

We know that 37=3×33×33=3×27×27=3(272
The number immediately before 27 that is divisible by 8 is 24.
Hence, replace 27 with 24+3.
Then we have:
37=3(272)=3(24+3)2=3(242+2×24×3+32
=3×242+3×2×24×3+3×9
Now,
37/8=(3×242+3×2×24×3+3×9)/8
=3×242/8+3×2×24×3/8+3×9/8
=Integer + Integer +27/8
=Integer + Integer+(24+3)/8
=Integer + Integer+3+3/8
Hence, the remainder is 3