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Q:

What is the remainder when $15^{23} + 23^{23}$ is divided by 19 ?

A) 4	B) 3
C) 1	D) 0

Answer & Explanation Answer: D) 0

Explanation:

$15^{23} + 23^{23} = {(19 - 4)}^{23} + {(19 + 4)}^{23}$

The remainder on dividing by 19 will be ${(- 4)}^{23} + {(4)}^{23}$ = 0.

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Q:

How many remainders are possible if $16^{n}$ is divided by 9 for any positive integral value of n ?

A) 1	B) 2
C) 3	D) 4

Answer & Explanation Answer: C) 3

Explanation:

When $16^{n}$ is divided by 9, we have

$\frac{16^{1}}{9}$ , remainder = 7

$\frac{16^{2}}{9}$ , remainder = 4

$\frac{16^{3}}{9}$ , remainder = 1

$\frac{16^{4}}{9}$ , remainder = 7

$\frac{16^{5}}{9}$ , remainder = 4

$\frac{16^{6}}{9}$ , remainder = 1

So, we have cyclicity of 3 factors i.e 7,4,1.

Hence only 3 remainders are possible.

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Q:

Find the unit digit of 4¹⁹⁸ + 6¹²³⁴⁵ + 348⁶⁶ + 24¹¹ + 1.

A) 3	B) 2
C) 1	D) 0

Answer & Explanation Answer: C) 1

Explanation:

Given that, $4^{198} + 6^{12345} + 348^{66} + 24^{11} + 1$

Now Power Cycle of

4 is 4, 6

6 is 6

8 is 8, 4, 2, 6

1 is 1

$4^{198}$ - 198 is even number , So the unit digit value is 6

$6^{12345}$ - It is always 6

$348^{66}$ - 66/4 the remainder is 2 , So the second value is 4

$24^{11}$ - 11 is odd number, So the unit digit value is 4

1 - It is always 1

Total = 21

Therefore, The unit digit is 1

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Q:

How many prime numbers exist in $6^{5} \times 35^{9} \times 11^{3}$ ?

A) 29	B) 31
C) 33	D) 27

Answer & Explanation Answer: B) 31

Explanation:

Given $6^{5} \times 35^{9} \times 11^{3}$

$\Rightarrow {(2 \times 3)}^{5} \times {(5 \times 7)}^{9} \times 11^{3}$

$\Rightarrow 2^{5} \times 3^{5} \times 5^{9} \times 7^{9} \times 11^{3}$

Thus, there are (5 + 5 + 9 + 9 + 3)= 31 prime numbers.

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Q:

If $3^{x - y} = 27$ , $3^{x + y} = 243$ then find the x value

A) 2	B) 8
C) 4	D) 1

Answer & Explanation Answer: C) 4

Explanation:

Given that $3^{x - y} = 27$ , $3^{x + y} = 243$

$\Rightarrow 3^{x - y} = 3^{3}, 3^{x + y} = 3^{5}$

=> x-y=3, x+y=5

=> x = 4

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Filed Under: Simplification
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Q:

${(31)}^{31} \times {(31)}^{- 27} = ?$

A) 961 x 861	B) 961 x 961
C) 961 x 2	D) 29791

Answer & Explanation Answer: B) 961 x 961

Explanation:

We know that $a^{m} \times a^{n} = a^{m + n}$

$∴ {(31)}^{31 - 27}$ = $31^{4}$ = 961 x 961.

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Q:

A,B and C enter into a partnership in the ratio $\frac{7}{2} : \frac{4}{3} : \frac{6}{5}$ . After 4 months, A increases his share by 50%. If the total profit at the end of one year be Rs. 21600, then B's share in the profit is :

A) Rs. 4000	B) Rs. 6000
C) Rs. 9000	D) Rs. 3000

Answer & Explanation Answer: A) Rs. 4000

Explanation:

Given ratio of initial investments = $\frac{7}{2} : \frac{4}{3} : \frac{6}{5}$ = 105 : 40 : 36.

Let the initial investments be 105x, 40x and 36x.

$∴ A : B : C = (105 x \times 4 + \frac{150}{100} \times 105 x \times 8) : (40 x \times 12) : (36 x \times 12)$

= 1680x : 480x : 432x = 35 : 10 : 9.

Hence, B's share = $(21600 \times \frac{10}{54})$ = Rs. 4000.

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Filed Under: Partnership
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Q:

A sales boy is allowed $4 \frac{1}{2}$ % discount on the total sales made by him plus a bonus of $\frac{1}{4}$ % on the sales over Rs.12,000.If his total earnings were Rs.2400, then his total sales(in Rs.) are ?

A) Rs.60,145.23	B) Rs.54,285.54
C) Rs.52,147.85	D) Rs.50,589.47

Answer & Explanation Answer: D) Rs.50,589.47

Explanation:

Let the total earnings be Rs.x

Then $4 \frac{1}{2}$ % of x + $\frac{1}{4}$ % of (x-12000) =2400

$\Rightarrow \frac{9}{2} \times \frac{x}{100} + \frac{1}{4} \times \frac{x - 12000}{100} = 2400$

=> x=50,589.47

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Filed Under: Percentage
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