If N = 6^6 + 6^7 + 6^8 + 6^9, what is the sum of the exponents of all prime factors in the… — GMAT

If $N = 6^6 + 6^7 + 6^8 + 6^9$ , what is the sum of the exponents of all prime factors in the prime factorization of $N$ ?

Factoring out

6^6

gives

N = 6^6(1 + 6 + 36 + 216) = 6^6(259)

. Decomposing composite bases into prime factors yields

6^6 = 2^6 \times 3^6

and

259 = 7^1 \times 37^1

. Thus, the prime factorization is

N = 2^6 \times 3^6 \times 7^1 \times 37^1

. Summing the prime exponents gives

6 + 6 + 1 + 1 = 14

Factor out the common term

6^6

from the expression.

N = 6^6(1 + 6 + 6^2 + 6^3) = 6^6(1 + 6 + 36 + 216) = 6^6 \times 259

Factoring simplifies the sum into a single product of terms.

Express

6^6

in terms of its prime factors.

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

The base 6 is composite and must be broken down into prime factors 2 and 3.

Determine the prime factorization of 259.

Testing small primes shows

259 = 7 \times 37

, where both 7 and 37 are prime numbers.

259 is not prime and must be decomposed into its prime components

7^1 \times 37^1

Combine all prime factors to write the complete prime factorization of

N

N = 2^6 \times 3^6 \times 7^1 \times 37^1

Writing

N

in standard canonical form reveals all prime exponents.

Sum the exponents of all prime factors.

6 + 6 + 1 + 1 = 14

The question asks for the total sum of the exponents of the prime factors.

Prime Factorization of Factored Exponential Sums

Estimated Time:2m 30s

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