What is the unit digit of $7^{2024}$?
Explanation: The unit digit of powers of 7 cycles every 4: 7, 9, 3, 1. Since 2024 ≡ 0 (mod 4), the unit digit is the last in the cycle: 1.
What is the unit digit of $1! + 2! + 3! + ... + 10!$?
Explanation: For n ≥ 5, n! ends in 0. So only 1!+2!+3!+4! = 1+2+6+24 = 33 contributes to the unit digit. The unit digit is 3.
How many trailing zeros are there in $50!$?
Explanation: Trailing zeros in n! come from factors of 5. Count = floor(50/5) + floor(50/25) = 10 + 2 = 12.
Which of the following statements about 91 is correct?
- 91 is composite because 91 = 7 × 13
- 91 is a prime number
- 91 is composite because 91 = 13 × 17
- 91 is divisible by 3
Explanation: 91 = 7 × 13, so it is composite. Many students mistakenly think 91 is prime because it is not divisible by 2, 3, or 5.
If p is a prime number greater than 3, then p² - 1 is always divisible by:
Explanation: For prime p > 3, p is odd so p-1 and p+1 are consecutive even numbers, giving a factor of 8. Also, one of p-1, p, p+1 is divisible by 3, and since p is prime > 3, either p-1 or p+1 is divisible by 3. Thus p²-1 is divisible by 8 × 3 = 24.
What is the HCF of 8 and 9?
Explanation: 8 = 2³ and 9 = 3². They share no common prime factors, so HCF(8,9) = 1. A common error is confusing HCF with LCM (72).
How many positive divisors does 72 have?
Explanation: 72 = 2³ × 3². Number of divisors = (3+1)(2+1) = 12. Common errors include counting only prime factors (2) or using 3 × 2 = 6.
What is the remainder when 16! is divided by 17?
Explanation: By Wilson's Theorem, for a prime p, (p-1)! ≡ -1 (mod p). Here p=17, so 16! ≡ -1 ≡ 16 (mod 17).
If p₁, p₂, ..., pₙ are all the prime numbers, then the number N = p₁p₂⋯pₙ + 1 is:
- Either prime or has a prime factor not in the original list
- Always prime
- Always composite
- Divisible by one of the pᵢ
Explanation: By Euclid's proof, N is not divisible by any pᵢ (remainder is 1). So N is either a new prime or has a prime factor not in the original list, proving primes are infinite.
According to Goldbach's conjecture, 100 can be expressed as the sum of two primes. Which pair is valid?
- 47 and 53
- 51 and 49
- 45 and 55
- 50 and 50
Explanation: 47 + 53 = 100 and both are prime. 51 = 3×17, 49 = 7², 45 = 9×5, 55 = 5×11, and 50 is not prime.
What is the unit digit of 2⁵¹ + 3⁵¹?
Explanation: Unit digit of 2⁵¹ is 8 (cycle: 2,4,8,6; 51 ≡ 3 mod 4). Unit digit of 3⁵¹ is 7 (cycle: 3,9,7,1; 51 ≡ 3 mod 4). Sum: 8+7=15, unit digit is 5.
What is the highest power of 3 that divides 50!?
Explanation: Count = floor(50/3) + floor(50/9) + floor(50/27) = 16 + 5 + 1 = 22.
What are the last two digits of 76²⁰²⁴?
Explanation: Numbers ending in 76 have the property that 76ⁿ always ends in 76 (automorphic number). This can be verified by binomial expansion: 76² = 5776.
Which statement about 161 is true?
- 161 is composite: 161 = 7 × 23
- 161 is prime
- 161 is divisible by 13
- 161 is a perfect square
Explanation: 161 = 7 × 23. It is not divisible by 13 (161 = 13 × 12 + 5) and 12² = 144, 13² = 169.
How many trailing zeros does 100! have?
Explanation: Count = floor(100/5) + floor(100/25) = 20 + 4 = 24.
What is the unit digit of 9²⁰²⁴?
Explanation: Powers of 9 cycle: 9, 1, 9, 1... For even exponents, unit digit is 1. 2024 is even, so unit digit is 1.
What is the HCF of 14 and 15?
Explanation: 14 = 2 × 7 and 15 = 3 × 5. They are consecutive integers, so HCF = 1. Common error: confusing with LCM = 210.
What is the sum of all positive divisors of 28?
Explanation: 28 = 2² × 7. Sum of divisors = (1+2+4)(1+7) = 7 × 8 = 56. 28 is a perfect number because sum of proper divisors = 28.
What is the remainder when 25! is divided by 29?
Explanation: By Wilson's theorem, 28! ≡ -1 (mod 29). Since 28! = 28×27×26×25! ≡ (-1)(-2)(-3)×25! = -6×25! (mod 29), we get -6×25! ≡ -1, so 6×25! ≡ 1. The inverse of 6 mod 29 is 5 (since 6×5 = 30 ≡ 1). Thus 25! ≡ 5 (mod 29).
What is the unit digit of 4²⁰²³?
Explanation: Powers of 4 cycle: 4, 6, 4, 6... For odd exponents, unit digit is 4. 2023 is odd, so unit digit is 4.
How many prime factors does 210 have (counted with multiplicity)?
Explanation: 210 = 2 × 3 × 5 × 7. All are prime, so there are exactly 4 prime factors.
What is the LCM of 8 and 12?
Explanation: 8 = 2³ and 12 = 2² × 3. LCM = 2³ × 3 = 24. Common error: HCF = 4.
What is the remainder when 6! is divided by 7?
Explanation: 6! = 720. 720 = 7 × 102 + 6. By Wilson's theorem, 6! ≡ -1 ≡ 6 (mod 7).
What is the unit digit of 3¹ + 3² + 3³ + ... + 3¹⁰?
Explanation: Unit digits of powers of 3 cycle: 3, 9, 7, 1. Sum of one full cycle = 20 (unit digit 0). Two full cycles (8 terms) give unit digit 0. Remaining terms: 3⁹ (unit 3) + 3¹⁰ (unit 9) = 12. Total unit digit is 2.
Which of the following numbers is prime?
Explanation: 87 = 3 × 29, 91 = 7 × 13, 93 = 3 × 31. Only 89 is prime.
How many positive divisors does 100 have?
Explanation: 100 = 2² × 5². Number of divisors = (2+1)(2+1) = 9.
What is the last non-zero digit of 7!?
Explanation: 7! = 5040. Removing the trailing zero, the last non-zero digit is 4.
What is the HCF of 16 and 25?
Explanation: 16 = 2⁴ and 25 = 5². They share no common prime factors, so HCF = 1.
What is the remainder when 10! is divided by 11?
Explanation: By Wilson's theorem, 10! ≡ -1 ≡ 10 (mod 11).
What is the unit digit of 5²⁰²⁴?
Explanation: Any positive power of 5 ends in 5. So 5²⁰²⁴ has unit digit 5.
How many trailing zeros does 75! have?
Explanation: Count = floor(75/5) + floor(75/25) = 15 + 3 = 18.
Which statement about 221 is correct?
- 221 is composite: 221 = 13 × 17
- 221 is prime
- 221 is divisible by 7
- 221 is a perfect cube
Explanation: 221 = 13 × 17. It is not divisible by 7 (221 = 7 × 31 + 4) and 6³ = 216, 7³ = 343.
What is the largest number that always divides p² - 1 for any prime p > 3?
Explanation: As proven earlier, for any prime p > 3, p² - 1 = (p-1)(p+1) is divisible by 8 (consecutive even numbers) and 3 (one of three consecutive integers). Thus divisible by 24.
Which of the following pairs is NOT co-prime?
- 14 and 21
- 8 and 9
- 14 and 15
- 21 and 22
Explanation: 14 = 2 × 7 and 21 = 3 × 7, so HCF(14,21) = 7. The other pairs are consecutive integers (HCF=1) or 8,9 (HCF=1).
How many positive divisors does 48 have?
Explanation: 48 = 2⁴ × 3¹. Number of divisors = (4+1)(1+1) = 10.
What is the unit digit of 8²⁰²⁴?
Explanation: Powers of 8 cycle: 8, 4, 2, 6. Since 2024 ≡ 0 (mod 4), the unit digit is 6.
What is the remainder when 4! is divided by 5?
Explanation: 4! = 24 = 5 × 4 + 4. Remainder is 4. Note: Wilson's theorem applies to (p-1)! for prime p, so 4! ≡ -1 ≡ 4 (mod 5).
What is the sum of the first 5 prime numbers?
Explanation: First 5 primes: 2, 3, 5, 7, 11. Sum = 28. Common error: omitting 2 gives 26, including 1 gives 29.
What is the highest power of 2 that divides 50!?
Explanation: Count = floor(50/2) + floor(50/4) + floor(50/8) + floor(50/16) + floor(50/32) = 25 + 12 + 6 + 3 + 1 = 47.
What is the unit digit of 1! × 2! × 3! × ... × 10!?
Explanation: Since 5! = 120 ends in 0, and all higher factorials also end in 0, the product contains a factor of 10. Thus the unit digit is 0.
Which pair sums to 50 and consists of two primes?
- 19 and 31
- 17 and 33
- 21 and 29
- 25 and 25
Explanation: 19 + 31 = 50 and both are prime. 33 = 3 × 11, 21 = 3 × 7, and 25 is not prime.
How many trailing zeros does 60! have?
Explanation: Count = floor(60/5) + floor(60/25) = 12 + 2 = 14.
Which statement about 143 is correct?
- 143 is composite: 143 = 11 × 13
- 143 is prime
- 143 is divisible by 7
- 143 is a perfect square
Explanation: 143 = 11 × 13. Not divisible by 7 (143 = 7 × 20 + 3) and 11² = 121, 12² = 144.
What is the LCM of 6 and 8?
Explanation: 6 = 2 × 3 and 8 = 2³. LCM = 2³ × 3 = 24. Common error: HCF = 2.
What is the unit digit of 6²⁰²⁴?
Explanation: Any positive power of 6 ends in 6. So 6²⁰²⁴ has unit digit 6.
How many positive divisors does 36 have?
Explanation: 36 = 2² × 3². Number of divisors = (2+1)(2+1) = 9.
What is the HCF of 18 and 25?
Explanation: 18 = 2 × 3² and 25 = 5². No common prime factors, so HCF = 1.
What is the remainder when 12! is divided by 13?
Explanation: By Wilson's theorem, 12! ≡ -1 ≡ 12 (mod 13).
What is the unit digit of 2¹ + 2² + 2³ + ... + 2¹⁰?
Explanation: Unit digits of powers of 2 cycle: 2, 4, 8, 6. Sum of one full cycle = 20 (unit digit 0). Two full cycles (8 terms) give unit digit 0. Remaining: 2⁹ (unit 2) + 2¹⁰ (unit 4) = 6. Total unit digit is 6.
How many prime factors does 30 have (counted with multiplicity)?
Explanation: 30 = 2 × 3 × 5. There are exactly 3 prime factors.
What is the highest power of 5 that divides 100!?
Explanation: Count = floor(100/5) + floor(100/25) = 20 + 4 = 24. This equals the number of trailing zeros.
Which statement about 119 is correct?
- 119 is composite: 119 = 7 × 17
- 119 is prime
- 119 is divisible by 13
- 119 is a perfect cube
Explanation: 119 = 7 × 17. Not divisible by 13 (119 = 13 × 9 + 2) and 4³ = 64, 5³ = 125.
What is the HCF of 15 and 16?
Explanation: 15 and 16 are consecutive integers, so they are always co-prime. HCF(15,16) = 1.
How many positive divisors does 60 have?
Explanation: 60 = 2² × 3¹ × 5¹. Number of divisors = (2+1)(1+1)(1+1) = 12.
What is the unit digit of 7! + 8! + 9! + 10!?
Explanation: 7! = 5040, 8! = 40320, 9! = 362880, 10! = 3628800. All end in 0, so their sum ends in 0.
What is the remainder when 22! is divided by 23?
Explanation: By Wilson's theorem, 22! ≡ -1 ≡ 22 (mod 23).
What is the unit digit of 3²⁰²⁴?
Explanation: Powers of 3 cycle: 3, 9, 7, 1. Since 2024 ≡ 0 (mod 4), the unit digit is 1.
What is the LCM of 9 and 12?
Explanation: 9 = 3² and 12 = 2² × 3. LCM = 2² × 3² = 36. Common error: HCF = 3.
How many trailing zeros does 125! have?
Explanation: Count = floor(125/5) + floor(125/25) + floor(125/125) = 25 + 5 + 1 = 31.
What is the sum of all positive divisors of 12?
Explanation: 12 = 2² × 3¹. Sum of divisors = (1+2+4)(1+3) = 7 × 4 = 28.