For advanced primary and middle school students. Take ten minutes, then explain why your answers work.
Problems
1. List all positive divisors of 24. Explain how you know your list is complete.
2. Two bells ring together now. One rings every 6 minutes and the other every 8 minutes. How many times do they ring together in the next 100 minutes? Exclude the starting time.
3. Find every positive integer n for which the greatest common divisor of n and 24 is 6 and their least common multiple is 72.
4. Find the smallest positive integer with exactly 12 positive divisors. Prove that it is the smallest.
Solutions
1. The factor pairs are (1, 24), (2, 12), (3, 8) and (4, 6). Every pair contains a factor at most √24, so checking 1 through 4 is enough.
2. Four times. The common interval is 24 minutes, so they ring together at minutes 24, 48, 72 and 96.
3. Only n = 18. The product of the GCD and LCM equals 24n, so n = 6 × 72 ÷ 24 = 18. Checking gives GCD 6 and LCM 72.
4. The answer is 60. For a prime factorization, the divisor count is the product of the exponents plus one. The multiplicative partitions of 12 are 12, 6 × 2, 4 × 3 and 3 × 2 × 2. For each pattern the smallest number uses the smallest primes, with larger exponents on smaller primes. The four minima are 2¹¹ = 2048, 2⁵ × 3 = 96, 2³ × 3² = 72 and 2² × 3 × 5 = 60.
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