GMAT Practice Question: The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n...
Question
The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example, the prime sum of 12 is 7 , since 12=2 \times 2 \times 3 and 2+2+3 =7. For which of the following integers is the prime sum greater than 35 ?
- 440
- 512
- 620
- 700
- 750
Topics: prime factorization, prime numbers, number theory
Solution
Step 1
We restate that the \textbf{\text("prime sum")} of a number is defined as the sum of its prime factors, counting repetitions.
\text{prime sum}(n) = \text{sum of prime factors of }n\text{ including repetitions}
Step 2
We factor 440 into its prime factors.
440 = 44 \times 10 44 = 4 \times 11 4 = 2 \times 2 10 = 2 \times 5 440 = 2 \times 2 \times 2 \times 5 \times 11
Step 3
We compute the prime sum for 440.
2 + 2 + 2 + 5 + 11 = 22
Step 4
We factor 512 into its prime factors using exponent notation.
512 = 2^9
Step 5
We compute the prime sum for 512.
9 \times 2 = 18
Step 6
We factor 620 into its prime factors.
620 = 62 \times 10 62 = 2 \times 31 10 = 2 \times 5 620 = 2 \times 2 \times 5 \times 31
Step 7
We compute the prime sum for 620.
2 + 2 + 5 + 31 = 40
Step 8
We factor 700 into its prime factors.
700 = 7 \times 100 100 = 10 \times 10 10 = 2 \times 5 700 = 2 \times 2 \times 5 \times 5 \times 7
Step 9
We compute the prime sum for 700.
2 + 2 + 5 + 5 + 7 = 21
Step 10
We factor 750 into its prime factors.
750 = 75 \times 10 75 = 15 \times 5 15 = 3 \times 5 10 = 2 \times 5 750 = 3 \times 5 \times 5 \times 2 \times 5
Step 11
We compute the prime sum for 750.
3 + 5 + 5 + 2 + 5 = 20
Step 12
We compare the prime sums and identify which exceeds 35.
440: 22 512: 18 620: 40 700: 21 750: 20 40 > 35
Answer
620
