**Direction To Solve**

**Choose the correct option**
**1-** 186 x 186 + 159 x 159 - 2 x 186 x 159 =?

Answer & Explanation

Answer - **C** (729)

Explanation - Given Exp.

*a*^{2} + b^{2} - 2ab, where *a* = 186 and *b* = 159

= (*a - b)*^{2} = (186 - 159)^{2} = (27)^{2}

= (20 + 7)^{2} = (20)^{2} + 7 + 2 x 20 x 7 = 400 + 49 + 280 = 729

**2-** The number of prime factors of (3 x 5)

^{12} (2 x 7)

^{10} (10)

^{25} is:

Answer & Explanation

Answer - **D** (94)

Explanation - (3 x 5)^{12} x (2 x 7)^{10} x (10)^{25} = (3 x 5)^{12} x (2 x 7)^{10} x (2 x 5)^{25}

= 3^{12} x 5^{12} x 2^{10} x 7^{10} x 2^{25} x 5^{25}

= 2^{35} x 3^{12} x 5^{37} x 7^{10}

Total number of prime factors = (35 + 12 + 37 + 10) = 94

**3-** If (64)

^{2} - (36)

^{2} = 20

*z*, the value of

*z* is:

Answer & Explanation

Answer - **A** (140)

Explanation - 20*z* = (64)^{2} - (36)^{2}

20*z* = (64 + 36) (64 - 36)

20*z* - 100 x 28

z = (100 x 28) / 20

= 140

**4-** Which of the following numbers is divisible by 3, 7, 9 and 11?

Answer & Explanation

Answer - **B** (2079)

Explanation - (a) 639 is not divisible by 7

(b) 2079 is divisible by 3, 7, 9 and 11

(c) 3791 is not divisible by 3

(d) 37911 is not divisible by 9

**5-** 39798 + 3798 + 378 =?

Answer & Explanation

Answer - **B** (43974)

Explanation - 39798 + 3798 + 378 = 43974

**6-** A six-digit number is formed by repeating a three-digit number; for example, 256256 or 678678 etc. Any number of this form is always exactly divisible by:

Answer & Explanation

Answer - **D** (1001)

Explanation - 256256 = 256 x 1001; 678678 = 678 x 1001, etc.

So, any number of this form is divisible by 1001

**7-** The unit's digit in the product (3127)

^{173} is:

Answer & Explanation

Answer - **C** (7)

Explanation - Unit digit in (3127)^{173} - Unit digit in (7)^{173}. Now, 7^{4} gives unit digit 1

(7)^{173} = (7^{4})^{43} x 7^{1}. Thus, (7)^{173} gives unit digit 7

**8-** The number of digits of the smallest number, which when multiplied by 7 gives the result consisting entirely of nines, is:

Answer & Explanation

Answer - **C** (6)

Explanation - By hit and trial, we find that a number exactly divisible by 7 and consisting entirely of nines is 999999.

Number of digits in it = 6

**9-** Which of the following numbers is exactly divisible by 99?

Answer & Explanation

Answer - **A** (114345)

Explanation - The required number should be divisible by both 9 and 11.

Clearly, 114345 is divisible by both 9 and 11. So, it is divisible by 99

**10-** On dividing a number by 999, the quotient is 366 and the remainder is 103. The number is:

Answer & Explanation

Answer - **C** (365737)

Explanation - Required number = 999 x 366 + 103 = (1000 - 1) x 366 + 103 = 366000 - 366 + 103

= 365737