Remainder MCQ Quiz - Objective Question with Answer for Remainder - Download Free PDF

Given:

p, q, r are positive integers.

When divided by 14, the remainders are:

p mod 14 = 5

q mod 14 = 8

r mod 14 = 9

Expression to evaluate: 2p + 3q - 3r mod 14

Formula Used:

If a mod n = x and b mod n = y , then:

(ka + mb) mod n = (k ⋅ x + m ⋅ y) mod n

Calculation:

Given:

p mod 14 = 5

q mod 14 = 8

r mod 14 = 9

Expression: 2p + 3q - 3r

Substitute the remainders.

⇒ 2p + 3q - 3r mod 14 = (2 ⋅ 5) + (3 ⋅ 8) - (3 ⋅ 9) mod 14

⇒ 10 + 24 - 27 mod 14

⇒ 7 mod 14

The remainder is 7.

Given:

Number = 21⁴²

Divisor = 441

Formula used:

If a number N can be expressed as k × m, where k is an integer and m is the divisor, then the remainder when N is divided by m is 0.

Calculations:

We are asked to find the remainder when 21⁴² is divided by 441.

We can write the divisor as 441 = 21 × 21.

Now, let's look at the number 21⁴².

21⁴² = 21 × 21 × 21 × ... × 21 (42 times)

We can group the first two terms:

21⁴² = (21 × 21) × (21 × 21 × ... × 21) (40 times)

Since 21 × 21 = 441,

21⁴² = 441 × (21⁴⁰)

Let k = 21⁴⁰. Since 21 is an integer, k is also an integer.

So, 21⁴² can be expressed in the form of k × 441, where k is an integer.

If a number can be expressed as a multiple of the divisor, the remainder is 0.

∴ The remainder when 21⁴² is divided by 441 is 0.

Given:

Find the remainder when 27²² is divided by 729.

Formula used:

If a number is raised to a power and divided by another number, modular arithmetic can be used: \((a^b \mod n)\).

Calculations:

27²² mod 729

729 = 27²

⇒ We simplify modulo 729 in terms of powers of 27.

⇒ 27²² = (27²)¹¹ × 27⁰

⇒ (27²) mod 729 = 0 (since 27² = 729)

⇒ Therefore, (27²² mod 729) = 0

∴ The correct answer is option (4).

Given:

We need to find the remainder when 23³² is divided by 529.

Concept Used:

Euler's Totient Theorem: If a and n are coprime, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.

For n = p^k where p is a prime number, φ(n) = p^k - p^k-1.

Calculation:

Calculate φ(529)

Since 529 = 23², we have φ(529) = 23² - 23¹ = 529 - 23 = 506.

Since 23 and 529 are not coprime, we cannot directly apply Euler's theorem. However, we can rewrite 23³² as (23²)¹⁶ = 529¹⁶.

We want to find the remainder of 529¹⁶ when divided by 529.

Since 529¹⁶ is a multiple of 529, the remainder is 0.

∴ The remainder when 23³² is divided by 529 is 0.

Every natural number is of the form 3k or 3k + 1 or 3k + 2 where k is a whole number.

(3k)², when divided by 9, will leave a remaidner of 0.

(3k + 1)² = 9k² + 6k + 1

k can be of the form 3k, or 3k₁ + 3k₁ + 1 or 3k₁ + 2 where k₁ is a whole number.

Therefore, 6k + 1 can be 18k₁ + 1 or 18k₁ + 7 or 18k₁ + 13.

Therefore, (3k + 1)² when divided by 9 leaves a remainder of 1 or 7 or 4.

Similarly, it can be shown (3k + 2)² leaves a remainder of 1 or 7 or 4.

Therefore, The square of a natural number leaves a remainder of 0 or 1 or 4 or 7. The sum of all the possible remainders is 12.

Given:

A number when divided by 7 leaves a remainder of 4. 

Concept used:

Dividend = Divisor × Quotient + Remainder

Calculation:

Let the number and quotient be N and D respectively.

​According to the question,

N = 7D + 4

⇒ N² = (7D + 4)²

⇒ N² = 49D² + 56D + 16

⇒ N² = 7(7D² + 8D + 2) + 2

So, 7(7D2 + 8D + 2) is completely divisible by 7.

Now, the remainder is 2.

∴ The remainder is 2.

Calculation:

8127

⇒ 8000 + 120 + 7

Here, both 8000 and 120 are divisible by 8.

Now, the remainder when 8127 is divided by 8 is 7

∴ The remainder when 8127 is divided by 8 is 7.

Given:

Find the remainder when 27²² is divided by 729.

Formula used:

If a number is raised to a power and divided by another number, modular arithmetic can be used: \((a^b \mod n)\).

Calculations:

27²² mod 729

729 = 27²

⇒ We simplify modulo 729 in terms of powers of 27.

⇒ 27²² = (27²)¹¹ × 27⁰

⇒ (27²) mod 729 = 0 (since 27² = 729)

⇒ Therefore, (27²² mod 729) = 0

∴ The correct answer is option (4).

Given:

We need to find the remainder when 23³² is divided by 529.

Concept Used:

Euler's Totient Theorem: If a and n are coprime, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.

For n = p^k where p is a prime number, φ(n) = p^k - p^k-1.

Calculation:

Calculate φ(529)

Since 529 = 23², we have φ(529) = 23² - 23¹ = 529 - 23 = 506.

Since 23 and 529 are not coprime, we cannot directly apply Euler's theorem. However, we can rewrite 23³² as (23²)¹⁶ = 529¹⁶.

We want to find the remainder of 529¹⁶ when divided by 529.

Since 529¹⁶ is a multiple of 529, the remainder is 0.

∴ The remainder when 23³² is divided by 529 is 0.

Given:

A number when divided by 7 leaves a remainder of 4. 

Concept used:

Dividend = Divisor × Quotient + Remainder

Calculation:

Let the number and quotient be N and D respectively.

​According to the question,

N = 7D + 4

⇒ N² = (7D + 4)²

⇒ N² = 49D² + 56D + 16

⇒ N² = 7(7D² + 8D + 2) + 2

So, 7(7D2 + 8D + 2) is completely divisible by 7.

Now, the remainder is 2.

∴ The remainder is 2.

Calculation:

8127

⇒ 8000 + 120 + 7

Here, both 8000 and 120 are divisible by 8.

Now, the remainder when 8127 is divided by 8 is 7

∴ The remainder when 8127 is divided by 8 is 7.

Given:

The remainder when x²⁵ + 1 is divided by x + 1.

Formula Used:

According to the Remainder Theorem, the remainder of a polynomial f(x) divided by x - c is f(c).

Calculation:

Here, f(x) = x²⁵ + 1 and we need to find the remainder when divided by x + 1, which is the same as finding f(-1).

f(x) = x²⁵ + 1

f(-1) = (-1)²⁵ + 1

f(-1) = -1 + 1

f(-1) = 0

The remainder when x²⁵ + 1 is divided by x + 1 is 0.

Given:

Number = 21⁴²

Divisor = 441

Formula used:

If a number N can be expressed as k × m, where k is an integer and m is the divisor, then the remainder when N is divided by m is 0.

Calculations:

We are asked to find the remainder when 21⁴² is divided by 441.

We can write the divisor as 441 = 21 × 21.

Now, let's look at the number 21⁴².

21⁴² = 21 × 21 × 21 × ... × 21 (42 times)

We can group the first two terms:

21⁴² = (21 × 21) × (21 × 21 × ... × 21) (40 times)

Since 21 × 21 = 441,

21⁴² = 441 × (21⁴⁰)

Let k = 21⁴⁰. Since 21 is an integer, k is also an integer.

So, 21⁴² can be expressed in the form of k × 441, where k is an integer.

If a number can be expressed as a multiple of the divisor, the remainder is 0.

∴ The remainder when 21⁴² is divided by 441 is 0.

Given:

Find the remainder when 27²² is divided by 729.

Formula used:

If a number is raised to a power and divided by another number, modular arithmetic can be used: \((a^b \mod n)\).

Calculations:

27²² mod 729

729 = 27²

⇒ We simplify modulo 729 in terms of powers of 27.

⇒ 27²² = (27²)¹¹ × 27⁰

⇒ (27²) mod 729 = 0 (since 27² = 729)

⇒ Therefore, (27²² mod 729) = 0

∴ The correct answer is option (4).

Given:

We need to find the remainder when 23³² is divided by 529.

Concept Used:

Euler's Totient Theorem: If a and n are coprime, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.

For n = p^k where p is a prime number, φ(n) = p^k - p^k-1.

Calculation:

Calculate φ(529)

Since 529 = 23², we have φ(529) = 23² - 23¹ = 529 - 23 = 506.

Since 23 and 529 are not coprime, we cannot directly apply Euler's theorem. However, we can rewrite 23³² as (23²)¹⁶ = 529¹⁶.

We want to find the remainder of 529¹⁶ when divided by 529.

Since 529¹⁶ is a multiple of 529, the remainder is 0.

∴ The remainder when 23³² is divided by 529 is 0.