**Division Algorithm:-**If we divide a number by another number, then

**Dividend = (Divisor * Quotient) + Remainder**

D = d * Q + R

Example :- 7 = 3 * 2 + 1

**Divisibility Rules**

**Divisibility by 2:-**

The last digit of the given number should be 'zero' or 'even'

Example:- 12, 2498760, 34562, 745974 is divisible by 2.

**Divisibility by 3:-**

The sum of the digits of the number should be divisible by 3.

Example:- 1.) 123 => 1 + 2 + 3 = 6 = 6/3 = 2

Example:- 2.) 72954 => 7 + 2 + 9 + 5 + 4 = 27/3 = 9

**Divisibility by 4:-**

The last two digit of the given number should be 'zero' or 'divisible by 4.

Example:- 1.) 65837348 is divisible by 4, last two digits 48 is divisible by 4

Example:- 2.) 09214900 is divisible by 4, last two digits 00.

**Divisibility by 5:-**

The last digit of the given number should be 'zero' or '5'.

Example:- 1.) 8421745 is divisible by 5,

Example:- 2.) 9497440 is divisible by 5.

**Divisibility by 6:-**

The given number should be divisible by both 2 and 3.

Example:- 1.) 47562 is divisible by 2 as well as 3. so it is divisible by 6.

Example:- 2.) 59676 is divisible by 2 as well as 3. so it is divisible by 6.

**Divisibility by 8:-**

The last three digit of the given number should be 'zero' or 'divisible by 8.

Example:- 1.) 5478924568 is divisible by 8, last three digits 568 is divisible by 8

Example:- 2.) 69214000 is divisible by 8, last two digits 000.

**Divisibility by 9:-**

The sum of the digits of the number should be divisible by 9.

Example:- 1.) 786546 => 7 + 8 + 6 + 5 + 4 + 6 = 36 = 36/9 = 4

Example:- 2.) 72 => 7 + 2 = 9/9 = 1

**Divisibility by 11:-**

A number is divisible by 11, if the difference of the sum of its digits at odd places

and the sum of its digits at even places is either 0 or a nmber divisible by 11.

Example:- 1331, 14641, 123123, 9141.

1.

On dividinng 109 by a number, the quotient is 9 and the remainder is 1. Find the divisor.

**Answer: Option 'B'**

**d = (D-R)/Q
= (109 - 1)/9
= 108/9 = 12**

2.

27^{20} / 27^{10} = 3^{k}, then k = ?

**Answer: Option 'B'**

**27 ^{20} / 27^{10} = 3^{k} => 27^{10} = 3^{k} => (3^{3})10 = 3^{k} => 3^{30} = 3^{k}.**

3.

On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder ?

**Answer: Option 'B'**

**Formula: (Divisor × Quotient) + Remainder = Dividend. Soln: (56 × Q)+29 = D -------(1) D%8 = R -------------(2) From equation(2), ((56 × Q)+29)%8 = R. => Assume Q = 1. => (56+29)%8 = R. => 85%8 = R => 5 = R.**

4.

5358 x 51 = ?

**Answer: Option 'A'**

**5358 x 51 = 5358 x (50 + 1) = 5358 x 50 + 5358 x 1 = 267900 + 5358 = 273258.**

5.

Find the HCF of 59 , 12?

**Answer: Option 'A'**

**The fastest way to find the correct answer is by elimination rule :
Check whether each of the given choices divides both the numbers. If any of the given choices doesnâ€™t divide both the numbers, then you can eliminate that particular choice and move to the next choice. Answer will be the HIGHEST Number that exactly divides both given numbers.
Please Note : There may be a scenario when two or more numbers in the choice will exactly divide both the numbers, in that case , please choose the Highest among the numbers.**

6.

The integers 32489 and 33714, when divided by a three digit number "n", leave the same remainder. What is the value of "n"?

**Answer: Option 'C'**

**Given two numbers 32489 and 33714.
Let r be the remainder we get, when we divide these both numbers by n, then
32489 = qn + r and 33714 = pn + r,
Where p and q are some integers.
Subtracting these two equations, we get
1225 = (p-q) x n.
We can write 1225 as 5 x 245.
Hence 245 is the answer.**

7.

What is the diffference between the place value and face value of 3 in the numeral 1375?

**Answer: Option 'C'**

**place value of 3 = 3 × 100 = 300
face value of 3 = 3
300 - 3 = 297**

8.

What is the unit digit in 3^{37}?

**Answer: Option 'A'**

**Cyclicity of 3 also = 4
D = d × Q + R
37 = 4 × 9 + 1
37/4 = Remainder 1, 31 = 3
Note:-
Cyclicity of 2, 3, 7, 8 => 4
Cyclicity of 4, 9 => 2
Cyclicity of 1, 5, 6 => 1**

9.

Find the Missing number in the the below series

117, 144, 171, X , 225, 252

**Answer: Option 'C'**

**The given series is -- 117, 144, 171, X , 225, 252
Consequtive numbers increases by 27
Following this pattern, the Missing Number X should increase by 27 from the previous number.
When we add 27 to the number before X (i.e 171) we get 171+27=198**

10.

The product of two consecutive even numbers is 15128. Find the average of these two numbers.

**Answer: Option 'A'**

**We wil go from the option. We will eliminate 133 and 139, since their product will exceed 16900.
Now the tug of war is between 123 and 127.
Lets take 123, if 123 is the average then the consecutive even numbers will be 122 and 124.
Now, observe, 122 x 124 = ....8, that is the resulting number ends with 8.
Now you try for 127.
(HINT : The consecutive even number with 127 as average us 126 and 128)
The two numbers are 122 and 124. Average is 123.**

11.

On dividing a number by 357, we get 39 as remainder. On dividing the same number by 17, what will be the remainder?

**Answer: Option 'C'**

**Let X be the number and y be the quotient. Then, X = 357 x y + 39 =(17x21xy)+(17x2)+5 =17 x (21y+2)+5 So, required number = 5**

12.

On multiplying a number by 7, the product is a number each of whose digits is 3. The smallest such number is:

**Answer: Option 'A'**

**By hit and trial, we find that 47619 x 7 = 333333.**

13.

The difference between a two-digit number and the number obtained by interchanging the digit is 27. Find the difference between the digits.

**Answer: Option 'C'**

**Let the tens and unit digits be x and y respectively.
Then, two digit number be 10x + y
The number obtained by interchanging the digit is 10y + x
**

14.

A number when divided by 216 leaves remainder 36 and the value of the quotient is 11. Find the number

**Answer: Option 'A'**

**Number = Divisor x Quotient + Remainder = 216 x 11 + 36 = 2376 + 36 = 2412**

15.

The difference between a two digit number and the number obtained by interchanging two digits of the number is 9.The sum of the two digits of the number is 15. What is product of the two digits of two digit number?

**Answer: Option 'B'**

**Sum of the digits is 15, this can happen only when the digits are 7 and 8.
The number is 87, interchanging digits we get 78,
Difference between these two numbers = 9
Product of these two digits = 7 x 8 = 56**

16.

A bell in a clock rings once at 1 O'clock, twice at 2 O'clock, thrice at 3 O'clock and so on.. Then how many times it rings in a day.

**Answer: Option 'D'**

**sum of n natural number = n(n+1)/2
= 12(13)/2 = 78
= 2 times (78) = 156**

17.

The difference between the local value and the face value of 7 in the numeral 32675149 is

**Answer: Option 'B'**

**(Local value of 7) - (Face value of 7) = (70000 - 7) = 69993**

18.

A boy multiplied 987 by a certain number and obtained 559981 as his answer. If in the answer both 9 are wrong and the other digits are correct, then the correct answer would be:

**Answer: Option 'A'**

**987 = 3 x 7 x 47 So, the required number must be divisible by each one of 3, 7, 47 553681 (Sum of digits = 28, not divisible by 3) 555181 (Sum of digits = 25, not divisible by 3) 555681 is divisible by 3, 7, 47.**

19.

The LCM of two numbers is 280 and their ratio is 7 : 8. The two numbers are

**Answer: Option 'D'**

**Let the number be 7x and 8x
HCF = x
We know HCF × LCM = product of two numbers
HCF × LCM = 7x × 8x
280x = 56x**

20.

Find the HCF of 19 , 8?

**Answer: Option 'A'**

**The fastest way to find the correct answer is by elimination rule :
Check whether each of the given choices divides both the numbers. If any of the given choices doesnâ€™t divide both the numbers, then you can eliminate that particular choice and move to the next choice. Answer will be the HIGHEST Number that exactly divides both given numbers.
Please Note : There may be a scenario when two or more numbers in the choice will exactly divide both the numbers, in that case , please choose the Highest among the numbers.**

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