# Problems on Numbers for Exam

Q21. Find the unit digit in the product (545 X 432 X 326)

(a) 0

(b) 1

(c) 2

(d) 10

**Answer:** (a) 0

Product of unit digits are

= 5 X 2 X 6

=60

∴ Required digit =0

Q22. The unit digit in (3^{65} X 6^{59} X 7^{71}) is

(a) 1

(b) 2

(c) 3

(d) 4

**Answer:** (d) 4

3^{65} = (3^{4})^{16} X 3

The unit digit of 3^{4} is 1

∴ The unit digit of (3^{4})^{16} is 1

∴ The unit digit of 3^{65} is 1 X 3 = 3

The unit digit of 6^{59} is 6

The unit digit of 7^{4} is 1. So unit digit in (7^{4})^{17} is 1

Now 7^{71} = (7^{4})^{17} X 7^{3} and unit digit in 7^{3} is 3

∴ unit digit in 7^{71} is 3

∴ Required unit digit = Unit digit in (3 X 6 X 3)

=unit digit in 54

=4

Q23. (45 + 46 + 47 + ⋅ ⋅ ⋅ ⋅ ⋅ + 113 + 114 + 115) = ?

(a) 5120

(b) 5570

(c) 5680

(d) 8650

**Answer:** (c) 5680

45 + 46 + 47 + ⋅ ⋅ ⋅ ⋅ ⋅ + 113 + 114 + 115

= (1 + 2 + ⋅ ⋅ ⋅ ⋅ ⋅ + 44 + 45 + 46 ⋅ ⋅ ⋅ ⋅ ⋅ + 115) - (1 + 2 + 3 + ⋅ ⋅ ⋅ ⋅ ⋅ + 44)

=115(115 + 1) | − | 44(44 + 1) |

2 | 2 |

115 X 116 | − | 44 X 45 |

2 | 2 |

=115 X 58 - 22 X 45

=6670-990

=5680

Formula 1 + 2 + ⋅ ⋅ ⋅ ⋅ + n = n(n+1) ÷ 2

Q24. A, B, C and D are four consecutive odd integers and their average is 32. What is the product of B and D ?

(a) 1085

(b) 1508

(c) 1805

(d) 1580

**Answer:** (a) 1085

Let the four integers are x+1, x+3, x+5, x+7

Nowx+1+x+3+x+5+x+7 | = | 32 |

4 |

=> | 4x + 16 | = | 32 |

4 |

=> 4x = (32 X 4) - 16

=> | x | = | 128 - 16 |

4 |

= | 112 |

4 |

=28

Now A = x+1 = 28+1 = 29

Now B = x+3 = 28+3 = 31

Now C = x+5 = 28+5 = 33

Now D = x+7 = 28+7 = 35

Product of B and D are 31 X 35 = 1085

Q25. A, B, C and D are four consecutive even integers and their average is 62. What is the product of A and C ?

(a) 3717

(b) 3771

(c) 3777

(d) None of the above

**Answer:** (a) 3717

Let A =x+2, B=x+4, C=x+6, D=x+8

Nowx+2+x+4+x+6+x+8 | = | 62 |

4 |

=> | 4x + 20 | = | 62 |

4 |

=> 4x = (62 X 4) - 20

=> | x | = | 248 - 20 |

4 |

= | 228 |

4 |

=57

A = 57 + 2 = 59

C = 57 + 6 = 63

A X C = 59 X 63 = 3717

Q26. 106 X 106 - 94 X 94 = ?

(a) 2000

(b) 2400

(c) 1904

(d) 1906

**Answer:** (b) 2400

106 X 106 - 94 X 94

=106^{2} - 94^{2}

=(106 + 94) (106 - 94)

=200 X 12

=2400

Q27.

5 | X | 2 | ÷ | 4 | ÷ | 6 |

6 | 9 | 9 | 7 |

(a) 0.44

(b) 0.32

(c) 0.49

(d) 0.36

**Answer:** (c) 0.49

5 | X | 2 | ÷ | 4 | ÷ | 6 |

6 | 9 | 9 | 7 |

5 | X | 2 | X | 9 | X | 7 |

6 | 9 | 4 | 6 |

35 |

72 |

=0.49

Q28. 8988 ÷ 8 ÷ 4 = ?

(a) 266.336

(b) 272.156

(c) 280.875

(d) 281.997

**Answer:** (c) 280.875

8988 ÷ 8 ÷ 4

=8988 X (1/8) X (1/4)

=8988 ÷ 32

=280.875

Q29. The sum and product of two numbers are 15 and 45 respectively. The sum of their reciprocals is

(a) 1/2

(b) 1/3

(c) 1/7

(d) 3/5

**Answer:** (b) 1/3

Let the numbers are a and b.

Now a+b = 15 and ab = 45

a+b | = | 15 |

ab | 45 |

a | + | b | = | 1 |

ab | ab | 3 |

1 | + | 1 | = | 1 |

b | a | 3 |

1 | + | 1 | = | 1 |

a | b | 3 |

Q30. The first prime number is

(a) 0

(b) 2

(c) 1

(d) 3

**Answer:** (b) 2