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Problem #60 · September 25, 2026

Prime numbers

For the digit nn, let 30+n30+n be a composite number. For which value of nn is the number of distinct prime divisors of this number the smallest?
  1. A44
  2. B22
  3. C55
  4. D11

WORKED EXAMPLE

Solution

We check the options: 30+4=34=2⋅1730+4=34=2\cdot17 and 30+5=35=5⋅730+5=35=5\cdot7 — two distinct prime factors each; 30+1=3130+1=31 is a prime number, so it does not satisfy the condition of being composite; 30+2=32=2530+2=32=2^5 has only one distinct prime factor (22). Therefore, the fewest distinct prime factors occur at n=2n=2.
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