What is the sum of all the distinct prime factors of the integer 6^8 - 6^6? — GMAT

What is the sum of all the distinct prime factors of the integer $6^8 - 6^6$ ?

Cevap: 17

The sum of the distinct prime factors of

6^8 - 6^6

is 17.

Factoring out

6^6

yields

6^6(6^2 - 1) = 6^6 \times 35

. The prime factors of

6^6

are 2 and 3, while the prime factors of 35 are 5 and 7. The distinct prime factors are 2, 3, 5, and 7. Summing these prime numbers gives

2 + 3 + 5 + 7 = 17

Factor out the common term

6^6

6^8 - 6^6 = 6^6(6^2 - 1) = 6^6 \times 35

Factoring simplifies the large exponent expression into a product of smaller integers.

Break down each base into its prime factors

6^6 = (2 \times 3)^6 = 2^6 \times 3^6

and

35 = 5 \times 7

Prime factorization requires expressing all bases as prime numbers.

List the distinct prime bases and compute their sum

Distinct prime factors: 2, 3, 5, 7. Sum =

2 + 3 + 5 + 7 = 17

Exponents do not affect which prime numbers are factors, only how many times they divide the number.

Prime Factorization of Difference of Exponents

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