According to question ,
Number = 269 × 68 + 0
Number = 269 × (67 + 1)

Correct Option: B

According to question ,
Number = 269 × 68 + 0
Number = 269 × (67 + 1)
Number = 269 × 67 + 269
Clearly, remainder is obtained on dividing 269 by 67 that is 1.
Thus , required remainder is 1.

Here, 357 is exactly divisible by 17.
∴  Required remainder = Remainder obtained on dividing 39 by
17

Correct Option: C

Here, 357 is exactly divisible by 17.
∴  Required remainder = Remainder obtained on dividing 39 by
17
⇒ 39 = ( 17 × 2 ) + 5
Hence required answer is 5 .

Using the Binomial expansion , we have
(x + 1)ⁿ = xⁿ + ⁿc₁ x^n–1 +
ⁿc₂ x^{n– 2} + ..... + ⁿc_n–1 x +1
Here, each term except the last term contains x. Obviously, each term except the last term is exactly divisible by x.

Correct Option: B

Using the Binomial expansion , we have
(x + 1)ⁿ = xⁿ + ⁿc₁ x^n–1 +
ⁿc₂ x^{n– 2} + ..... + ⁿc_n–1 x +1
Here, each term except the last term contains x. Obviously, each term except the last term is exactly divisible by x.
Following the same logic,
∴ 7¹⁹ = (6 + 1)¹⁹ has each term except last term divisible by 6.
Hence, 7¹⁹ + 2 when divided by 6 leaves remainder
⇒ 7¹⁹ + 2 = 1 + 2 = 3
Hence the remainder is 3.

On dividing the given number by 119, let k be the quotient and 19 as remainder.
Then, number = 119k + 19

Correct Option: D

On dividing the given number by 119, let k be the quotient and 19 as remainder.
Then, number = 119k + 19
number = 17 × 7k + 17 × 1 + 2
number = 17 (7k + 1) + 2
Hence, the given number when divided by 17, gives (7k + 1) as quotient and 2 as remainder.
Hence , required remainder is 2.

Let the numbers be p and q and p is greater than q.
As given,
The product of two numbers = 9375
pq = 9375     .......(i)
Again,

Correct Option: C

Let the numbers be p and q and p is greater than q.
As given,
The product of two numbers = 9375
pq = 9375     .......(i)
Again,