# Numbers

### Important Formulas

Some Basic Formulae:

1. (a + b)(ab) = (a2b2)

2. (a + b)2 = (a2 + b2 + 2ab)

3. (ab)2 = (a2 + b2 – 2ab)

4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

5. (a3 + b3) = (a + b)(a2ab + b2)

6. (a3b3) = (ab)(a2 + ab + b2)

7. (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2abbcac)

8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

1

Which one of the following is not a prime number?
 A. 31 B. 61 C. 71 D. 91

Explanation:

91 is divisible by 7. So, it is not a prime number.

2.

(112 x 54) = ?
 A. 67000 B. 70000 C. 76500 D. 77200

Explanation:

 (112 x 54) = 112 x 10 4 = 112 x 104 = 1120000 = 70000 2 24 16

3.

It is being given that (232 + 1) is completely divisible by a whole number. Which of the following numbers is completely divisible by this number?
 A. (216 + 1) B. (216 – 1) C. (7 x 223) D. (296 + 1)

Explanation:

Let 232 = x. Then, (232 + 1) = (x + 1).

Let (x + 1) be completely divisible by the natural number N. Then,

(296 + 1) = [(232)3 + 1] = (x3 + 1) = (x + 1)(x2x + 1), which is completely divisible by N, since (x + 1) is divisible by N.

4.

What least number must be added to 1056, so that the sum is completely divisible by 23 ?
 A. 2 B. 3 C. 18 D. 21 E. None of these

Explanation:

``` 23) 1056 (45
92
---
136
115
---
21
---

Required number = (23 - 21)
= 2.   ```

5.

1397 x 1397 = ?
 A. 1951609 B. 1981709 C. 18362619 D. 2031719 E. None of these

Explanation:

 1397 x 1397 = (1397)2 = (1400 – 3)2 = (1400)2 + (3)2 – (2 x 1400 x 3) = 1960000 + 9 – 8400 = 1960009 – 8400 = 1951609.

6.

How many of the following numbers are divisible by 132 ?
264, 396, 462, 792, 968, 2178, 5184, 6336
 A. 4 B. 5 C. 6 D. 7

Explanation:

132 = 4 x 3 x 11

So, if the number divisible by all the three number 4, 3 and 11, then the number is divisible by 132 also.

264 11,3,4 (/)

396 11,3,4 (/)

462 11,3 (X)

792 11,3,4 (/)

968 11,4 (X)

2178 11,3 (X)

5184 3,4 (X)

6336 11,3,4 (/)

Therefore the following numbers are divisible by 132 : 264, 396, 792 and 6336.

Required number of number = 4.

7.

The largest 4 digit number exactly divisible by 88 is:
 A. 9944 B. 9768 C. 9988 D. 8888 E. None of these

Explanation:

```Largest 4-digit number = 9999

88) 9999 (113
88
----
119
88
----
319
264
---
55
---

Required number = (9999 - 55)
= 9944.        ```

8.

What is the unit digit in {(6374)1793 x (625)317 x (341491)}?
 A. 0 B. 2 C. 3 D. 5

Explanation:

Unit digit in (6374)1793 = Unit digit in (4)1793

= Unit digit in [(42)896 x 4]

= Unit digit in (6 x 4) = 4

Unit digit in (625)317 = Unit digit in (5)317 = 5

Unit digit in (341)491 = Unit digit in (1)491 = 1

Required digit = Unit digit in (4 x 5 x 1) = 0.

9.

The sum of first five prime numbers is:
 A. 11 B. 18 C. 26 D. 28

Explanation:

Required sum = (2 + 3 + 5 + 7 + 11) = 28.

Note: 1 is not a prime number.

Definition: A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.

10.

The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number ?
 A. 240 B. 270 C. 295 D. 360

Explanation:

Let the smaller number be x. Then larger number = (x + 1365). x + 1365 = 6x + 15 5x = 1350 x = 270 Smaller number = 270.

11

If the number 517*324 is completely divisible by 3, then the smallest whole number in the place of * will be:
 A. 0 B. 1 C. 2 D. None of these

Explanation:

Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3. x = 2.

12.

The smallest 3 digit prime number is:
 A. 101 B. 103 C. 109 D. 113

Explanation:

The smallest 3-digit number is 100, which is divisible by 2. 100 is not a prime number.

101 < 11 and 101 is not divisible by any of the prime numbers 2, 3, 5, 7, 11. 101 is a prime number.

Hence 101 is the smallest 3-digit prime number.

13.

Which one of the following numbers is exactly divisible by 11?
 A. 235641 B. 245642 C. 315624 D. 415624

Explanation:

(4 + 5 + 2) – (1 + 6 + 3) = 1, not divisible by 11.

(2 + 6 + 4) – (4 + 5 + 2) = 1, not divisible by 11.

(4 + 6 + 1) – (2 + 5 + 3) = 1, not divisible by 11.

(4 + 6 + 1) – (2 + 5 + 4) = 0, So, 415624 is divisible by 11.

14.

(?) – 19657 – 33994 = 9999
 A. 63650 B. 53760 C. 59640 D. 61560 E. None of these

Explanation:

``` 19657         Let x - 53651  = 9999
33994         Then, x = 9999 + 53651 = 63650
-----
53651
-----
```

15.

The sum of first 45 natural numbers is:
 A. 1035 B. 1280 C. 2070 D. 2140

Explanation:

Let Sn =(1 + 2 + 3 + … + 45). This is an A.P. in which a =1, d =1, n = 45.

 Sn = n [2a + (n – 1)d] = 45 x [2 x 1 + (45 – 1) x 1] = 45 x 46 = (45 x 23) 2 2 2

= 45 x (20 + 3)

= 45 x 20 + 45 x 3

= 900 + 135

= 1035.

Shorcut Method:

 Sn = n(n + 1) = 45(45 + 1) = 1035. 2 2