We can see the denominator of 1^st is cancelled with a numerator of 2^nd and this process is repeated

So, After cancelling, we get

⇒ 2/n

Given:

Numbers to be taken = 1 to 78

Concept Used:

Sum of n terms of a series = n(n + 1)/2

Calculation:

Number Series for sum = 1 + 2 + 3 + ... + 78

Here, n = 78

Putting the value in the formula,

Sum of series = 78 (78 + 1) / 2 = 39 × 79

Hence, the sum of the first 78 natural numbers is divisible by 79, as it is one of the factors.

Given:

HCF = 24

LCM = 168

Ratio of numbers = 1 ∶ 7.

Formula:

Product of numbers = LCM × HCF

Calculation:

Let numbers be x and 7x.

x × 7x = 24 × 168

⇒ x² = 24 × 24

⇒ x = 24

∴ Larger number = 7x = 24 × 7 = 168.

GIVEN:

2³⁸⁴ is divided by 17.

CALCULATION:

2³⁸⁴ = 2^{(4 × 96)} = 16⁹⁶

We know that when 16 is divided by 17 the remainder is -1

When 16⁹⁶ is divided by 17 then remainder = (-1)⁹⁶ = 1.

Given:

The H.C.F. of (x³ + x² + x + 1) and (x⁴ – 1) is

Calculation:

⇒  (x³ + x² + x + 1) = x²(x + 1) + 1(x + 1)

⇒ (x + 1) (x² + 1)

⇒ x⁴ – 1 = (x² – 1) (x² + 1)

⇒ (x + 1) (x – 1) (x² + 1)

∴ Required HCF is (x + 1) (x² + 1)

Given:

27²⁷ + 27

Concept used:

Aⁿ + Bⁿ is divisible by (A + B) when n is odd.

Calculation:

Now, (2727 + 27)

⇒ (27²⁷ + 1²⁷ + 27 - 1)

⇒ (27²⁷ + 1²⁷) + 26

Here, according to the concept, (27²⁷ + 1²⁷) is divisible by (27 + 1) i.e. 28.

Hence, the remainder = 26

∴ The remainder when 27²⁷ + 27 is divided by 28 is 26.

Given:

A six-digit number Is divisible by 33

Formula used:

Dividend = divisor × quotient + remainder

Calculation:

Dividend = divisor × quotient + remainder

⇒ 33 × q + 0 = 33q

If 54 is added to the dividend then,

New number = 33q + 54

⇒ 3 × (11q + 18)

So, we can clearly say that the new number is divisible by 3.

∴ The correct option is 1.

Mistake Points

Please note that this is the official paper of SSC and SSC has given the 3 as the correct answer, but 111111 is also the 6 digit number and if we add 54 it will be divisible by both 3 and 5.

GIVEN:

LCM of two numbers is 189 and the numbers are in the ratio 9 : 7.

CONCEPT:

LCM: It is a number that is a multiple of two or more numbers.

CALCULATION:

Suppose the numbers are 9x and 7x.

⇒ LCM of 9x and 7x = 9 × 7 × x = 63x

According to the question,

63x = 189

⇒ x = 3

∴ Sum of the numbers = 9x + 7x = 16x = 48

Given:

The 8-digit number 123456xy is divisible by 8

Concept used:

If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Calculation:

So, 6xy should be divisible by 8

Now,

Possible numbers are 600, 608, 616, 624, 632, 640, 648, 656, 664, 672, 680, 688, 696

So, total of 13 possible pairs can be made

∴ The required answer is 13.