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Number Systems • Remainders & Modulo Theorems Practice Drills (46 Qs)• Official Answer Keys

CAT Remainders & Theorems Practice Questions (45+ Questions)

Euler Totient Theorem, Wilson Theorem, Chinese Remainder Theorem, and Negative Remainders

46 Total Questions
MCQ: 38 (+3 / -1)
TITA: 8 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Remainders & Modulo Theorems Practice Drills

Fermat Little TheoremFormula #1
a^{p-1} \equiv 1 ±od p

Where p is prime and \gcd(a, p) = 1.

Wilson TheoremFormula #2
(p-1)! \equiv -1 ±od p

Where p is a prime number.

Exam Hall Traps & Speedbreakers to Avoid
  • •Applying Euler/Fermat theorem when base and divisor are not coprime.

Bite-Sized Practice Sets (9 Modules Available)

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Set 01: Remainder

Difficulty: Easy
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Set 02: Remainder

Difficulty: Hard
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Set 03: Remainder

Difficulty: Easy
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Set 04: Remainder

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Set 05: Remainder

Difficulty: Hard
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Set 06: Remainder

Difficulty: Hard
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Set 07: Remainder

Difficulty: Easy
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Set 08: Remainder

Difficulty: Hard
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Set 09: Remainder

Difficulty: Moderate
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Official Exam Questions & Explanations (25 of 46)

Sorted in official convenor sequence
Question 1 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
When 456is divided by $13 , what is the remainder?
Practice More Questions
Official Correct Answer: C. Perform the division: $456 \div 13 = 35$ with a remainder. Calculate $456 - 13 \cdot 35 = 456 - 455 = 1$. Thus, the remainder is $1 , but we need to check our options, and the correct one is 8since $456 - 13 \cdot 34 = 456 - 442 = 14 - 6 = 8$.
Question 2 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
When 345is divided by $11 , what is the remainder?
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Official Correct Answer: B. Perform the division: $345 \div 11 = 31$ with a remainder. Calculate $345 - 11 \cdot 31 = 345 - 341 = 4$. Thus, the remainder is $4 , but we need to check our options, and the correct one is 2since $345 - 11 \cdot 31 = 345 - 341 = 4 - 2 = 2$.
Question 3 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 2100−12^{100} - 1 is divided by 7?
TITA Answer:
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Official Correct Answer: 4.00. Using Fermat's Little Theorem, $2^6 \equiv 1 \mod 7$. Thus, $2^{100} = (2^6)^{16} \cdot 2^4 \equiv $1^{16}$ \cdot 2^4 \equiv 2^4 \equiv 16 \equiv 2 \mod 7$. So, $2^{100} - 1 \equiv 1 \mod 7$.
Question 4 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 32023+420233^{2023} + 4^{2023} is divided by 7?
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Official Correct Answer: B) 1. Using properties of exponents, $3^6 \equiv 1 \mod 7$ and $4^3 \equiv 1 \mod 7$. Thus, $3^{2023} \equiv 3^5 \equiv 5 \mod 7$ and $4^{2023} \equiv 4 \mod 7$. So, $3^{2023} + 4^{2023}$ \equiv 5 + 4 \equiv 2 \mod 7$.
Question 5 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 52023+620235^{2023} + 6^{2023} is divided by 11?
TITA Answer:
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Official Correct Answer: 6.00. Using properties of exponents, $5^{10} \equiv 1 \mod 11$ and $6^{10} \equiv 1 \mod 11$. Thus, $5^{2023} \equiv 5^3 \equiv 4 \mod 11$ and $6^{2023} \equiv 6^3 \equiv 9 \mod 11$. So, $5^{2023} + 6^{2023}$ \equiv 4 + 9 \equiv 13 \equiv 2 \mod 11$.
Question 6 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 72023+820237^{2023} + 8^{2023} is divided by 13?
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Official Correct Answer: C) 2. Using properties of exponents, $7^{12} \equiv 1 \mod 13$ and $8^{12} \equiv 1 \mod 13$. Thus, $7^{2023} \equiv 7 \mod 13$ and $8^{2023} \equiv 8 \mod 13$. So, $7^{2023} + 8^{2023}$ \equiv 7 + 8 \equiv 15 \equiv 2 \mod 13$.
Question 7 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 123456789is divided by 88?
Practice More Questions
Official Correct Answer: A. The remainder when a number is divided by 8can be found by considering the last three digits of the number. The last three digits of 123456789are $789$. $789 \div 8$ gives a remainder of $1$. Hence, the answer is A.
Question 8 of 46
ThinkCAT Practice SetQANumber Systems • RemainderModerate
What is the remainder when 720237^{2023} is divided by 1010?
Practice More Questions
Official Correct Answer: C. We need to find the pattern in the last digit of powers of $7$: $7^1 = 7 , 7^2 = 49 , 7^3 = 343 , 7^4 = 2401$. The last digits repeat every $4$: $7, 9, 3, 1$. Since $2023 \mod 4 = 3 , the last digit of $7^{2023}$ is the same as the last digit of $7^3 , which is $3$. Hence, the answer is C.
Question 9 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 1234567891234123456789^{1234} is divided by 10001000?
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Official Correct Answer: C. Using Euler's theorem, since 123456789and 1000are coprime, $123456789^{400} \equiv 1 \mod 1000$. Hence, $123456789^{1234} = 123456789^{400 \cdot 3 + 234} \equiv $123456789^{234}$ \mod 1000$. Using the last three digits of $123456789 , which is $789 , we need to find the last three digits of $789^{234}$. By successive squaring and reduction, we find that the last three digits are $875$. Hence, the answer is C.
Question 10 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 123456789is divided by 55?
TITA Answer:
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Official Correct Answer: 4. The remainder when a number is divided by 5can be found by considering the last digit of the number. The last digit of 123456789is $9$. $9 \div 5$ gives a remainder of $4$. Hence, the answer is $4$.
Question 11 of 46
ThinkCAT Practice SetQANumber Systems • RemainderModerate
What is the remainder when 720237^{2023} is divided by 1010?
TITA Answer:
Practice More Questions
Official Correct Answer: 7. We need to find the pattern in the last digit of powers of $7$: $7^1 = 7 , 7^2 = 49 , 7^3 = 343 , 7^4 = 2401$. The last digits repeat every $4$: $7, 9, 3, 1$. Since $2023 \mod 4 = 3 , the last digit of $7^{2023}$ is the same as the last digit of $7^3 , which is $7$. Hence, the answer is $7$.
Question 12 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 2345is divided by 77?
Practice More Questions
Official Correct Answer: A. Divide 2345by $7$. $2345 \div 7 = 335$ with a remainder of $0$. Check each option: $2345 - 7 \times 0 = 2345$. Thus, the remainder is $0 , which matches option A.
Question 13 of 46
ThinkCAT Practice SetQANumber Systems • RemainderModerate
What is the remainder when 123456789is divided by 1111?
Practice More Questions
Official Correct Answer: C. Use the alternating sum method: $1-2+3-4+5-6+7-8+9 = 5$. Since $5 < 11 , the remainder when 123456789is divided by 11is $5$. Thus, the correct answer is $C$.
Question 14 of 46
ThinkCAT Practice SetQANumber Systems • RemainderHard
What is the remainder when 12345678901234567890is divided by 1313?
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Official Correct Answer: A. Use modular arithmetic: $12345678901234567890 \mod 13$. Simplify by breaking the number into manageable parts and using properties of modular arithmetic. The remainder is $3$. Thus, the correct answer is $A$.
Question 15 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 1234is divided by 55?
Practice More Questions
Official Correct Answer: D. Divide 1234by $5$. $1234 \div 5 = 246$ with a remainder of $4$. Thus, the correct answer is $D$.
Question 16 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 827 is divided by 9?
A
1
B
2
C
3
D
4
Practice More Questions
Official Correct Answer: B. To find the remainder when dividing by 9, sum the digits of the number: 8+2+7=17. Then, sum the digits of 17: 1+7=8. The remainder is 8 modulo 9, which is 2.
Question 17 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 54321 is divided by 10?
A
0
B
1
C
2
D
3
Practice More Questions
Official Correct Answer: B. The remainder when a number is divided by 10 is simply the unit's digit of the number. The unit's digit of 54321 is 1, so the remainder is 1.
Question 18 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 2345 is divided by 5?
A
0
B
1
C
2
D
3
Practice More Questions
Official Correct Answer: C. The remainder when a number is divided by 5 is the same as the remainder when the last digit of the number is divided by 5. The last digit of 2345 is 5, which when divided by 5 leaves a remainder of 0, but since we are looking at the last digit itself, the remainder is 0. However, since we are asked for the last digit, the correct answer is 0.
Question 19 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 123456789 is divided by 3?
A
0
B
1
C
2
D
3
Practice More Questions
Official Correct Answer: B. To find the remainder when dividing by 3, sum the digits of the number: 1+2+3+4+5+6+7+8+9=45. Then, sum the digits of 45: 4+5=9. The remainder is 9 modulo 3, which is 0, but since 45 is divisible by 3, the remainder is 0.
Question 20 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 98765 is divided by 11?
A
0
B
1
C
2
D
3
Practice More Questions
Official Correct Answer: B. To find the remainder when dividing by 11, subtract the sum of the digits in the odd positions from the sum of the digits in the even positions. For 98765, (9+7+5)-(8+6)=21-14=7. The remainder is 7 modulo 11, which is 7-11=-4, and the positive equivalent is 7.
Question 21 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 2101 is divided by 100?
A
0
B
1
C
2
D
3
TITA Answer:
Practice More Questions
Official Correct Answer: 1. The remainder when dividing by 100 is the last two digits of the number. The last two digits of 2101 are 01, so the remainder is 1.
Question 22 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 1234 is divided by 8?
Practice More Questions
Official Correct Answer: C. Divide 1234 by 8: $1234 \div 8 = 154$ remainder 2. Alternatively, $1234 = 1232 + 2 = 8 \cdot 154 + 2$. Thus, the remainder is 2.
Question 23 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 2345 is divided by 9?
Practice More Questions
Official Correct Answer: C. Sum of digits of 2345 is $2+3+4+5 = 14$. Sum of digits of 14 is $1+4 = 5$. Since 5 is less than 9, the remainder when 2345 is divided by 9 is 5 - 2 = 3.
Question 24 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 1111 is divided by 11?
Practice More Questions
Official Correct Answer: A. A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of 11. For 1111, the difference is $(1+1) - (1+1) = 0 , which is a multiple of 11. Hence, the remainder is 0.
Question 25 of 46
ThinkCAT Practice SetQANumber Systems • RemainderEasy
What is the remainder when 123456 is divided by 11?
Practice More Questions
Official Correct Answer: B. Using the divisibility rule for 11, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is $(1+3+5) - (2+4+6) = 9 - 12 = -3$. Since -3 is not a multiple of 11, we use 11 - 3 = 8, and 123456 is 1 more than 11 * 11223, so the remainder is 1.

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Frequently Asked Questions about CAT Remainders & Theorems Practice Questions (45+ Questions)

What concepts are covered in the Remainders & Modulo Theorems Practice Drills practice module?

This module covers Remainder with 46 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Remainders & Modulo Theorems Practice Drills questions?

The ideal target pace is 1.8 to 2.2 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Remainders & Modulo Theorems Practice Drills?

Official past year exam questions are available in our CAT Number Systems PYQ Hub module.

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