42

Nice. Do you want to work with the number 42 in a math problem, or are you just testing? Here are quick math facts and then a few practice prompts you can choose from. Key facts about 42 - Prime factorization: 42 = 2 × 3 × 7 - Number of positive divisors: d(42) = 8 (divisors: 1, 2, 3, 6, 7, 14, 21, 42) - Sum of divisors: σ(42) = 96 - Sum of proper divisors: 54 (so 42 is an abundant number) - Euler’s totient: φ(42) = 12 - Harshad (Niven) in base 10: 42 ÷ (4+2) = 7 - 42 is a 3-almost prime (it has three prime factors counting multiplicity) Practice options (you can pick one, or I can give more) 1) Divisors and pairs: List all divisors and verify that complementary pairs multiply to 42: (1,42), (2,21), (3,14), (6,7). 2) Totient and sigma: Show why φ(42) = 12 and σ(42) = 96 from the prime factorization. 3) Prove 42 is abundant: Sum the proper divisors and show it exceeds 42. 4) 3-almost prime check: Explain why 42 is a 3-almost prime (and give examples of other 3-almost primes). 5) Quick application: If n is chosen uniformly from 1 to 100, what is the probability that gcd(n,42) = 1? (Use φ(42) for the density, then approximate for the range.) If you tell me which option you want (or your subject), I’ll work it out step by step.