**Ans . **

(3) 8

**Explanation :**\(a^2-b^2 = (a^2+b^2)(a+b)(a-b)\)

Let a = 3, b = 1

Required number

= (3 + 1) (3 – 1) = 8

**Ans . **

(1) 4

**Explanation :**82. Let m = n = p and m – n = 2p m + n = 2p

(m – n) (m + n) = 4p2

**Ans . **

(3) 6

**Explanation :**83. A number is divisible by 9, if sum of its digits is divisible by 9.

Let the number be x.

5 + 4 + 3 + 2 + x + 7 = 21 + x

x = 6

**Ans . **

(1) 5

**Explanation :**84.A number is divisible by 9 if the sum of its digits is divisible by 9.

Here, 6 + 7 + 0 + 9 = 22

Now, 22 + 5 = 27, which is divis- ible by 9. Hence 5 must be add- ed to 6709.

**Ans . **

(2) 6

**Explanation :**A number is divisible by 9 and 6 both, if it is divisible by LCM of 9 and 6 i.e., 18. Hence, the numbers are 108, 126, 144, 162, 180, 198.

**Ans . **

(2) 150

**Explanation :**86.First 3–digit number divisible by 6 = 102

Last such 3-digit number =996

996 = 102 + (n –1) 6

(n – 1)6 = 996 – 102 = 894

n – 1 = \(\frac{894}{6}=149\)

n = 150

**Ans . **

(2) 6

**Explanation :**\( n^3 – n = n (n^2 – 1)\)

= n (n + 1) (n – 1)

For n = 2, n3 – n = 6

**Ans . **

(3) 6

**Explanation :**\( n^3 – n = n (n + 1) (n – 1)\)

\(n = 4, n^3 – n = 4 × 5 × 3 = 60\)

60 ÷ 6 = 10

**Ans . **

(3)divisible by 9

**Explanation :**Number = 100x + 10y + z

Sum of digits = x + y + z

Difference = 100x + 10y + z – x – y – z

= 99x + 9y = 9 (11x + y)

**Ans . **

divisible by (11 × 13)

**Explanation :**divisible by (11 × 13)

**Ans . **

(3) 5

**Explanation :**91.Any number is divisible by 11 when the differences of alterna- tive digits is 0 or multiple of 0, 11 etc. Here,

5 + 2 + * = 7 + *

8 + 4 = 12

* = 12 – 7 = 5

**Ans . **

(4) 5

**Explanation :**92.A number is divisible by 11, if the difference of the sum of its dig- its at odd places and the sum of its digits of even places, is either 0 or a number divisible by 11.

(5 + 9 + * + 7) – (4 + 3 + 8) = 0

or multiple of 11

21 + * – 15

* + 6 = a multiple of 11

* = 5

**Ans . **

(4) 1

**Explanation :**93. A number is divisible by 11, if the difference of sum of its dig- its at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

Difference = (4 + 3 + 7 + 8) – (2 + 8 + ) = 22 – 10 – * = 12 – *

Clearly, * = 1

**Ans . **

(4) 4

**Explanation :**94. A number is divisible by 11 if the difference of the sum of digits at odd and even places be either zero or multiple of 11.

If the middle digit be 4, then 24442 or 244442 etc are divisi- ble by 11.

**Ans . **

(2) 12

**Explanation :**\(n^2(n^2–1) = n^2 (n + 1) (n – 1)\)

Now, we put values n = 2, 3..... When n = 2

\(n^2(n^2 –1)\) = 4 × 3 × 1 = 12, which is a multiple of 12

When n = 3. n\(n^2(n^2 –1)\) = 9 × 4 × 2 = 72,

which is also a multiple of 12. etc

**Ans . **

(4)Smallest 3-digit prime num- ber

**Explanation :**Let the unit digit be x and ten’s digit be y.

Number = 1000y + 100x + 10y + x = 1010y + 101x = 101(10y + x)

Clearly, this number is divisible by 101, which is the smallest three-digit prime number.

**Ans . **

(2) 10004

**Explanation :**The least number of 5 digits = 10000

Required number = 10000 + (41–37)

= 10004

**Ans . **

(1) \(2^{96} + 1\)<

**Explanation :**\(2^{96} + 1 = (2^{32})^3 + 1^3\)

Clearly, \(2^{32}+1\) is a factor of \(2^{96} +1\)

**Ans . **

(4) 48

**Explanation :**For n = 2

\(n^4 + 6n^3 + 11n^2 + 6n + 24\)= 16 + 48 + 44 + 12 + 24 = 144

which is divisible by 48. Clearly, 48 is the required num- ber. = 144

which is divisible by 48. Clearly, 48 is the required num- ber.

**Ans . **

(2)18

**Explanation :**When we divide 1000 by 225, quotient = 4

When we divide 5000 by 225, quotient = 22

Required answer = 22 – 4 = 18

**Ans . **

(3) 24

**Explanation :**\((n^3 – n) (n – 2) = n (n – 1) (n + 1) (n – 2)\)

When n = 3, Number = 3 × 2 × 4 = 24

**Ans . **

(1) 1440

**Explanation :**LCM of 16 and 18 = 144 Multiple of 144 that is less than 1500 = 1440

**Ans . **

(2)6

**Explanation :**The largest 4-digit number = 9999

Required number = 345 – 339 = 6

**Ans . **

(2) 10

**Explanation :**\(4^{61} + 4^{62} + 4^{63} + 4^{64} = 4^{61}*(1 + 4 + 16 + 64) = 4^{61} * 85\)

Which is a multiple of 10.

**Ans . **

(2) 9

**Explanation :**Let the number be 10x + y

where y < x. Number obtained by interchang- ing the digits = 10y + x

Difference = 10x + y – 10y – x

= 9x – 9y = 9 (x – y)

Hence, the difference is always exactly divisible by 9.

**Ans . **

(3) 303375

**Explanation :**Check through option

\(\frac{303375}{25} = \frac{303375*4}{25*4} = 12135\)

A number is divisible by 25 if the last two digits are divisible by 25 or zero.

**Ans . **

(1) 132

**Explanation :**307 × 32 = 9824

307 × 33 = 10131

Required number = 10131 – 9999 = 132

**Ans . **

(1) 15999879

**Explanation :**a = 4011, b = 3989

ab = 4011 × 3989

= (4000 + 11) (4000 – 11) = (4000)2 – (11)2

= 16000000 – 121 = 15999879

**Ans . **

(2) 2

**Explanation :**Expression =\( 3^{2n} + 9^n + 5 \) = \( 3^{2n} + 9^n + 3 \) + 2

Clearly, remainder = 2

**Ans . **

(3) 7

**Explanation :**Resulting number = 3957 + 5349 – 7062 = 2244 which is divisible by 4, 3 and 11.

2244 ÷ 4 = 561

2244 ÷ 3 = 748

2244 ÷ 11 = 204

**Ans . **

(3) 7387

**Explanation :**Prime numbers between 80 and 90.

= 83 and 89

Required product = 83 × 89 = 7387

**Ans . **

(2) 35

**Explanation :**When n = 2, \(6^n – 1 = 6^2 – 1 = 36 – 1 = 35\)

When, n = an even number,

\(a^n–b^n\) is always divisible by \((a^2–b^2)\).

**Ans . **

(2) 5

**Explanation :**Total number of marbles = x

+ x + 3 + x – 3 = 3x

3x = 15 x = 5

**Ans . **

(3) 10 kg

**Explanation :**Bucket + full water = 17 kg.

Bucket + 1/2 water = 13.5 kg.

Water = 2 × 3.5 = 7 kg.

Weight of empty bucket 122.

= 17 – 7 = 10 kg.

**Ans . **

(4) 30

**Explanation :**A cow and a hen each has a head.

If the total number of cows be x,

then Number of hens = 180 – x

A cow has four legs and a hen has two legs.

(180 – x) × 2 + 4x = 420

360 – 2x + 4x = 420

2x = 420 – 360 = 60

**Ans . **

(4) 6

**Explanation :**On putting n = 1

n(n +1) (n + 2) = 1 × 2 × 3 = 6

**Ans . **

(2) 3

**Explanation :**2736 ÷ 24 = 114

Hence, first divisor (2736) is a multiple of second divisor (24).

Required remainder = Remainder obtained on dividing 75 by 24 = 3

**Ans . **

(2) 9

**Explanation :**5 E9 + 2 F8 + 3 G7 = 1114

Value of ‘F’ will be maximum if the values of E and G are mini- mum.

509 + 2 F8 + 307 = 1114

2 F8 = 1114 – 509 – 307= 298

F = 9

**Ans . **

(4)6, 10, 14, 18

**Explanation :**Let four numbers be a, b, c and d respectively.

a + b + c + d = 48 (i) and,

a + 5 = b + 1 = c – 3 = d – 7 = x (let)

a = x – 5; b = x – 1, c = x + 3, d = x + 7 From equation (i),

x – 5 + x – 1 + x + 3 + x + 7 = 48 4x + 4 = 48

4x = 48 – 4 = 44

x = 44 = 11

a = x – 5 = 11 – 5 = 6

b = x – 1 = 11 – 1 = 10

c = x + 3 = 11 + 3 = 14

d = x + 7 = 11 + 7 = 18

**Ans . **

(2) 24

**Explanation :**\(\frac{2055}{27}\)

Remainder is 3

Required number = 27 – 3 = 24

**Ans . **

\(\frac{n+1}{2}\)

**Explanation :**122.Sum of first n bers = \(\frac{n*(n+1)}{2}\)

Required avg. = \(\frac{n*(n+1)}{2*n}\) = \(\frac{(n+1)}{2}\)

**Ans . **

(3) 9

**Explanation :**123.Here, the first divisor (361) is a multiple of second divisor (19).

Required remainder = Re- mainder obtained on dividing 47 by 19 = 9

**Ans . **

(3) 990

**Explanation :**Largest number = 3995

Smallest number = 3005

Difference = 3995 – 3005 = 990

**Ans . **

(2) 1250

**Explanation :**Let the numbers be x and y. According to the question,

x + y = 75

x – y = 25

Q (x + y)2 – (x – y)2 = 4xy

752 – 252 = 4xy

4xy = (75 + 25) (75 – 25)

4xy = 100 × 50

xy = 1250

**Ans . **

(4) 95

**Explanation :**Required difference = 97 – 2 = 95

**Ans . **

(4) 10

**Explanation :**) xy = 24

(x, y)

= (1 × 24), (2 ×12), (3 × 8), (4 × 6)

Minimum value of (x + y) = 4 + 6 = 10.

**Ans . **

(3)Both 3 and 9

**Explanation :**128. Let the 3–digit number be 100x + 10y + z.

Sum of the digits = x + y + z According to the question, Difference

= 100x + 10y + z – (x + y + z)

= 99x + 9y = 9 (11x + y)

Clearly, it is a multiple of 3 and 9.

**Ans . **

(1) 60

**Explanation :**129. Let the numbers be x and y

where x > y.

According to the question, (x + y) – (x – y) = 30

x + y – x + y = 30

2y = 30

y = 30/2 = 15

xy = 900

15x = 900

x = 60

**Ans . **

(3) 5336

**Explanation :**130. According to the question, Divisor (d) = 5r = 5 × 46 = 230 Again, Divisor (d) = 10 × Quo- tient (q)

230 = q × 10

q = 230/10 = 23

Dividend = Divisor × Quotient + Remainder

= 230 × 23 + 46

= 5290 + 46 = 5336

**Ans . **

(3) 5

**Explanation :**Divided = 44 × 432 = 19008

19008/31 Remainder = 5

**Ans . **

(2) 2

**Explanation :**Here, first divisor (729) is a multiple of second divisor (27).

Required remainder = Remainder got on dividing 56 by 27 = 2.

**Ans . **

(4) 100008

**Explanation :**Smallest number of six dig- its = 100000

108/100000 remainder = 100

Required number = 100000 + (108 – 100) = 100008

**Ans . **

(2) 14

**Explanation :**Let the number be x. According to the question, x + 25 = 3x – 3

3x – x = 25 + 3

2x = 28

x = 14

**Ans . **

(1) 2

**Explanation :**334 × 545 × 7p is divisible by 3340.

334 × 5 × 109 × 7 × p, is

divisible by 334 × 2 × 5

Clearly, p = 2

**Ans . **

(2) 1

**Explanation :**Let the number be a. According to the question,

a + 1/a = 2

a2 + 1 = 2a a2 – 2a + 1 = 0

(a – 1)2 = 0 a – 1 = 0

a = 1

**Ans . **

(3) 5

**Explanation :**First divisor (56) is a mul- tiple of second divisor (8).

Required remainder = Remainder obtained after divid- ing 29 by 8 = 5

**Ans . **

(2) 7

**Explanation :**Let the number be x. According to the question,

x – 4 = 21/x

x2 – 4x = 21

x2 – 4x – 21 = 0

(x + 3) (x – 7) = 0

x = 7 because x is not equal to – 3.

**Ans . **

(2) 2

**Explanation :**Let quotient be 1.

n = 4 × 1 + 3 = 7

2n = 2 × 7 = 14,

On dividing 14 by 4, remainder = 2

**Ans . **

paste_right_option

**Explanation :**Divisor = 555 + 445 = 1000

Quotient = (555 – 445) × 2 = 110 × 2 = 220

Remainder = 30

Dividend = Divisor × Quotient + Remainder

= 1000 × 220 + 30 = 220030

**Ans . **

(1) 6480

**Explanation :**According to the question, Divisor = 2 × remainder = 2 × 80 = 160

Again, 4 × quotient = 160

Quotient = 160/4 = 40

x = Divisor × Quotient + re- mainder = 160 × 40 + 80 = 6480

**Ans . **

(2) 11

**Explanation :**Here, first divisor (342) is a multiple of second divisor (18). i.e. 342 ÷ 18 = 19

Required remainder = Remainder on dividing 47 by 18 = 11

**Ans . **

(3) 42

**Explanation :**Let second number = x. First number = 3x

Third number = 2/3 × 3x = 2x

According to the question, 3x + x + 2x = 252

6x = 252

x = 252/6 = 42

**Ans . **

(3) 65952

**Explanation :**Five-digit numbers formed by 2, 5, 0, 6 and 8 :

Largest number = 86520

Smallest number = 20568

Required difference = 86520 – 20568 = 65952

**Ans . **

(1) 21

**Explanation :**Let the number of cows be x. Q A hen or a cow has only one head.

Number of hens = 50 – x

A hen has two feet. A cow has four feet.

According to the question, 4x + 2 (50 – x) = 142

4x + 100 – 2x = 142

2x = 142 – 100 = 42

x = 21

**Ans . **

(2) 1683

**Explanation :**Firstly, we find LCM of 5, 6, 7 and 8.

LCM = 2 × 5 × 4 × 3 × 7= 840

Required number = 840x + 3 which is exactly divis- ible by 9.

Now, 840x + 3 = 93x × 9 + 3x + 3

When x = 2 then 840x + 3, is di- visible by 9.

Required number = 840 × 2 + 3 = 1683

**Ans . **

(4) 9

**Explanation :**A 3–digit number = 100x + 10y + z

Sum of digits = x + y + z

Difference = 100x + 10y + z – x – y – z

= 99x + 9y = 9 (11x + y)

i.e., multiple of 9.

**Ans . **

(1) 27

**Explanation :**8961/84 Remainder is 57

Required number = 84 – 57 = 27

**Ans . **

(1) 27

**Explanation :**Number of numbers lying be- tween 67 and 101

101 – 67 – 1 = 33

Prime numbers 71, 73, 79, 83, 89 and 97 = 6

Composite numbers = 33 – 6 = 27

**Ans . **

(3) 1

**Explanation :**LCM of 9, 11 and 13 = 9 × 11 × 13 = 1287

Required lowest number that leaves 6 as remainder = 1287 + 6 = 1293

Required answer = 1294 – 1293 = 1

**Ans . **

paste_right_option

**Explanation :**A number is divisible by 8 if number formed by the last three

If * is replaced by 3, then 632

**Ans . **

(4) 69

**Explanation :**13851/87 Remainder = 18

Required no. = 87 - 18 = 69

**Ans . **

(2) 8

**Explanation :**If the sum of the digits of a number be divisible by 9, the number is divisible by 9.

Sum of the digits of 451 * 603

= 4 + 5 + 1 + * + 6 + 0 + 3 = 19 + * If * = 8, then 19 + 8 = 27 which is divisible by 9.

**Ans . **

(2) 9944

**Explanation :**The largest 4-digit number = 9999

9999/88 Remainder is 55 Required number = 9999 – 55 = 9944

**Ans . **

(1) 57717

**Explanation :**= 5 + 7 + 7 + 1 + 7 = 27 which is divisible by 9

Required number = 57717

**Ans . **

(3) 187

**Explanation :**Prime numbers between 58 and 68 59, 61 and 67

Required sum = 59 + 61 + 67 = 187

**Ans . **

(3) 38

**Explanation :**Let the two digit number be 10x + y.

According to the question,

xy = 24 (i)

and, 10x + y + 45 = 10y + x

y - x = 5 ....(ii)

x + y = 11 ....(iii)

On adding equations (ii) and (iii),

y – x + x + y = 5 + 11

2y = 16 y = 8

xy = 24 8x = 24

x = 3

Required number = 10x + y = 10 × 3 + 8 = 38

**Ans . **

(3) 3

**Explanation :**A number is divisible by 11 if the difference between the sum of digits at odd places and that at even places is either zero or a multiple of 11.

Sum of the digits at odd places = 6 + 8 + 5 = 19

Sum of the digits at even places = 9 + 6 + 7 = 22

Required number=22–19 = 3

**Ans . **

(3) 12

**Explanation :**According to the question

\(\frac{2+2*5}{3} = 4\)

Second No. \(\frac{48}{4} = 12\)