The number of distinct prime factors of the largest 6-digit number is

Concept

The number of distinct prime factors of an integer is the count of unique prime bases in its full factorization — a repeated base (one raised to a power) is counted once, not once per exponent. The identity a2 − b2 = (a − b)(a + b) is useful for splitting a large number of the form 10n − 1 into two smaller factors.

Application

1. The largest 6-digit number is 106 − 1 = 999999.

2. Apply the difference-of-squares identity: 106 − 1 = (103 − 1)(103 + 1) = 999 × 1001.

3. Factor the first block: 999 = 27 × 37 = 33 × 37.

4. Factor the second block: 1001 = 7 × 11 × 13.

5. Combine both blocks: 999999 = 33 × 7 × 11 × 13 × 37.

6. Count only the unique prime bases (the exponent 3 on the base 3 does not add extra distinct primes): 3, 7, 11, 13, 37 — five distinct prime bases in total.

Cross-check

Multiplying the five bases back together confirms the factorization: 33 × 7 = 189, 189 × 11 = 2079, 2079 × 13 = 27027, 27027 × 37 = 999999 — matching the original number exactly, so no prime base was missed or added.

Therefore, the largest 6-digit number, 999999, has exactly 5 distinct prime factors: 3, 7, 11, 13, and 37.