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Since there are an infinite number of prime numbers, there are infinite numbers with any given number of prime factors.
Just multiply the first five prime numbers! 2x3x5x7x11 = 6x35x11 = 210x11 = 2310. Any multiple of this number "has the first five prime numbers as factors", but 2310 is the smallest such number.
All composite numbers can be expressed as unique products of prime numbers. This is accomplished by dividing the original number and its factors by prime numbers until all the factors are prime. A factor tree can help you visualize this. Example: 210 210 Divide by two. 105,2 Divide by three. 35,3,2 Divide by five. 7,5,3,2 Stop. All the factors are prime. 2 x 3 x 5 x 7 = 210 That's the prime factorization of 210.
Multiply prime squares by themselves. 24, 34, 54 all have five factors.
105, 210, 315, 420, 525