Remainder MCQ Quiz - Objective Question with Answer for Remainder - Download Free PDF
Last updated on Jun 11, 2025
Latest Remainder MCQ Objective Questions
Remainder Question 1:
Let p, q, r be positive integers such that when p, q, r are divided by 14, the remainders are 5, 8, 9 respectively. What will be the remainder when 2p + 3q – 3r is divided by 14 ?
Answer (Detailed Solution Below)
Remainder Question 1 Detailed Solution
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.
Remainder Question 2:
Find the remainder when 21^42 divided by 441.
Answer (Detailed Solution Below)
Remainder Question 2 Detailed Solution
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.
Remainder Question 3:
Find the remainder when 2722 divided by 729
Answer (Detailed Solution Below)
Remainder Question 3 Detailed Solution
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).
Remainder Question 4:
Find the remainder when 2332 divided by 529.
Answer (Detailed Solution Below)
Remainder Question 4 Detailed Solution
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 aφ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.
For n = pk where p is a prime number, φ(n) = pk - pk-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.
Remainder Question 5:
Find the sum of all the possible distinct remainders which can be obtained when the square of a natural number is divided by 9?
Answer (Detailed Solution Below)
Remainder Question 5 Detailed Solution
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.
Top Remainder MCQ Objective Questions
A number when divided by 7 leaves remainder of 4. If the square of the same number is divided by 7, then what is the remainder?
Answer (Detailed Solution Below)
Remainder Question 6 Detailed Solution
Download Solution PDFGiven:
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.
What is the remainder when 8127 is divided by 8?
Answer (Detailed Solution Below)
Remainder Question 7 Detailed Solution
Download Solution PDFCalculation:
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.
Find the remainder when 2722 divided by 729
Answer (Detailed Solution Below)
Remainder Question 8 Detailed Solution
Download Solution PDFGiven:
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).
Find the remainder when 2332 divided by 529.
Answer (Detailed Solution Below)
Remainder Question 9 Detailed Solution
Download Solution PDFGiven:
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 aφ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.
For n = pk where p is a prime number, φ(n) = pk - pk-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.
Remainder Question 10:
A number when divided by 7 leaves remainder of 4. If the square of the same number is divided by 7, then what is the remainder?
Answer (Detailed Solution Below)
Remainder Question 10 Detailed Solution
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.
Remainder Question 11:
What is the remainder when 8127 is divided by 8?
Answer (Detailed Solution Below)
Remainder Question 11 Detailed Solution
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.
Remainder Question 12:
The remainder when x25 + 1 is divided by x + 1 is
Answer (Detailed Solution Below)
Remainder Question 12 Detailed Solution
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.
Remainder Question 13:
Find the remainder when 21^42 divided by 441.
Answer (Detailed Solution Below)
Remainder Question 13 Detailed Solution
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.
Remainder Question 14:
Find the remainder when 2722 divided by 729
Answer (Detailed Solution Below)
Remainder Question 14 Detailed Solution
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).
Remainder Question 15:
Find the remainder when 2332 divided by 529.
Answer (Detailed Solution Below)
Remainder Question 15 Detailed Solution
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 aφ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.
For n = pk where p is a prime number, φ(n) = pk - pk-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.