Depth · one concept from first principles · step 2 of 4
Find the largest three-digit natural number that satisfies both conditions below.
(a) Its prime factorization has exactly different prime factors, and these prime factors add up to .
(b) It has exactly divisors.
Three odd primes add to an odd number, so 12 needs the prime 2. The other two are different primes adding to 10: 3 and 7.
The prime factors must be 2, 3, 7.
4.OA.B.4Eliminate PossibilitiesWrite N = 2^a × 3^b × 7^c with a, b, c at least 1. Each factor a+1, b+1, c+1 is at least 2, and their product is 12.
Two exponents are 1 and one exponent is 2.
6.EE.A.1Identify SubproblemsA number has divisors.
Every divisor is built only from the number's own primes , , , each used no more times than in the number, and two different builds never give the same divisor.
The number of s can be chosen ways ( up to ), the s ways and the s ways, each choice free of the others, so there are builds.
Put the exponent 2 on each prime in turn.
The largest three-digit one is 294.
6.EE.A.1Make A Systematic ListThe sum picks the primes; the divisor count picks the exponents.
A parent dashboard for the family lives at sensimlab.com.