Let the least number be x.

Correct Option: D

Let the least number be x.

y = 5 × 1 + 3 = 8
x = 13 × 8 + 1 = 105
On dividing 105 by 65,
⇒ 105 = ( 65 × 1 ) + 40
Hence required remainder = 40

Given , Remainder = 28
Let the quotient be Q and the remainder be R. Then
According to question ,
Divisor = 7 Q = 3 R

Correct Option: D

Given , Remainder = 28
Let the quotient be Q and the remainder be R. Then
According to question ,
Divisor = 7 Q = 3 R

Let the number be 10a + b .
After interchanging the digits, the number obtained = 10b + a.

Correct Option: A

Let the number be 10a + b .
After interchanging the digits, the number obtained = 10b + a.
According to the question,
Resulting number = 10a + b + 10b + a
Resulting number = 11a + 11b
Resulting number = 11 (a + b)
which is exactly divisible by 11.

Let two numbers are a and b and the divisor is d .
According to question ,

Correct Option: C

Let two numbers are a and b and the divisor is d .
According to question ,

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.