Numbers

Question:
Which of the following numbers will completely divide (325 + 326 + 327 + 328) ?

[A].

11

[B].

16

[C].

25

[D].

30

Answer: Option D

Explanation:

(325 + 326 + 327 + 328) = 325 x (1 + 3 + 32 + 33) = 325 x 40

     = 324 x 3 x 4 x 10

     = (324 x 4 x 30), which is divisible by30.

Question:
Which of the following numbers will completely divide (461 + 462 + 463 + 464) ?

[A].

3

[B].

10

[C].

11

[D].

13

Answer: Option B

Explanation:

(461 + 462 + 463 + 464) = 461 x (1 + 4 + 42 + 43) = 461 x 85

     = 460 x (4 x 85)

     = (460 x 340), which is divisible by 10.

Question: 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)

Answer: Option D

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)(x2 – x + 1), which is completely divisible by N, since (x + 1) is divisible by N.

Question: Which one of the following is the common factor of (4743 + 4343) and (4747 + 4347) ?
[A].

(47 – 43)

[B].

(47 + 43)

[C].

(4743 + 4343)

[D].

None of these

Answer: Option B

Explanation:

When n is odd, (xn + an) is always divisible by (x + a).

Each one of (4743 + 4343) and (4747 + 4347) is divisible by (47 + 43).

Question: 8597 – ? = 7429 – 4358

[A].

5426

[B].

5706

[C].

5526

[D].

5476

Answer: Option C

Explanation:

7429 Let 8597 – x = 3071
-4358 Then, x = 8597 – 3071
—- = 5526
3071
—-

Question:
What will be remainder when 17200 is divided by 18 ?

[A].

17

[B].

16

[C].

1

[D].

2

Answer: Option C

Explanation:

When n is even. (xn – an) is completely divisibly by (x + a)

(17200 – 1200) is completely divisible by (17 + 1), i.e., 18.

   (17200 – 1) is completely divisible by 18.

   On dividing 17200 by 18, we get 1 as remainder.

Question:
(xn – an) is completely divisible by (x – a), when

[A].

n is any natural number

[B].

n is an even natural number

[C].

n is and odd natural number

[D].

n is prime

Answer: Option A

Explanation:

For every natural number n, (xn – an) is completely divisible by (x – a).

Question: Which of the following numbers will completely divide (4915 – 1) ?
[A].

8

[B].

14

[C].

46

[D].

50

Answer: Option A

Explanation:

(xn – 1) will be divisibly by (x + 1) only when n is even.

(4915 – 1) = {(72)15 – 1} = (730 – 1), which is divisible by (7 +1), i.e., 8.

Question:
What will be remainder when (6767 + 67) is divided by 68 ?

[A].

1

[B].

63

[C].

66

[D].

67

Answer: Option C

Explanation:

(xn + 1) will be divisible by (x + 1) only when n is odd.

(6767 + 1) will be divisible by (67 + 1)

(6767 + 1) + 66, when divided by 68 will give 66 as remainder.

Question:
n is a whole number which when divided by 4 gives 3 as remainder. What will be the remainder when 2n is divided by 4 ?

[A].

3

[B].

2

[C].

1

[D].

0

Answer: Option B

Explanation:

Let n = 4q + 3. Then 2n = 8q + 6   = 4(2q + 1 ) + 2.

Thus, when 2n is divided by 4, the remainder is 2.