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 2 3 and 2+2+3 =7. For which of the following integers is the prime sum greater than 35 ?

GMAT Panda · Accepted Answer

620 We restate that the ("prime sum") of a number is defined as the sum of its prime factors, counting repetitions. prime sum(n) = sum of prime factors of n including repetitions We factor 440 into its prime factors. 440 = 44 10 44 = 4 11 4 = 2 2 10 = 2 5 440 = 2 2 2 5 11 We compute the prime sum for 440. 2 + 2 + 2 + 5 + 11 = 22 We factor 512 into its prime factors using exponent notation. 512 = 2^9 We compute the prime sum for 512. 9 2 = 18 We factor 620 into its prime factors. 620 = 62 10 62 = 2 31 10 = 2 5 620 = 2 2 5 31 We compute the prime sum for 620. 2 + 2 + 5 + 31 = 40 We factor 700 into its prime factors. 700 = 7 100 100 = 10 10 10 = 2 5 700 = 2 2 5 5 7 We compute the prime sum for 700. 2 + 2 + 5 + 5 + 7 = 21 We factor 750 into its prime factors. 750 = 75 10 75 = 15 5 15 = 3 5 10 = 2 5 750 = 3 5 5 2 5 We compute the prime sum for 750. 3 + 5 + 5 + 2 + 5 = 20 We compare the prime sums and identify which exceeds 35. 440: 22 512: 18 620: 40 700: 21 750: 20 40 > 35

Explanation	Calculations	Help
$We restate that the ("prime sum") of a number is defined as the sum of its prime factors, counting repetitions.$	$prime sum (n) = sum of prime factors of n including repetitions$
$We factor 440 into its prime factors.$	$440 = 44 \times 10 44 = 4 \times 11 4 = 2 \times 2 10 = 2 \times 5 440 = 2 \times 2 \times 2 \times 5 \times 11$
$We compute the prime sum for 440.$	$2 + 2 + 2 + 5 + 11 = 22$
$We factor 512 into its prime factors using exponent notation.$	$512 = 2^{9}$
$We compute the prime sum for 512.$	$9 \times 2 = 18$
$We factor 620 into its prime factors.$	$620 = 62 \times 10 62 = 2 \times 31 10 = 2 \times 5 620 = 2 \times 2 \times 5 \times 31$
$We compute the prime sum for 620.$	$2 + 2 + 5 + 31 = 40$
$We factor 700 into its prime factors.$	$700 = 7 \times 100 100 = 10 \times 10 10 = 2 \times 5 700 = 2 \times 2 \times 5 \times 5 \times 7$
$We compute the prime sum for 700.$	$2 + 2 + 5 + 5 + 7 = 21$
$We factor 750 into its prime factors.$	$750 = 75 \times 10 75 = 15 \times 5 15 = 3 \times 5 10 = 2 \times 5 750 = 3 \times 5 \times 5 \times 2 \times 5$
$We compute the prime sum for 750.$	$3 + 5 + 5 + 2 + 5 = 20$
$We compare the prime sums and identify which exceeds 35.$	$440 : 22 512 : 18 620 : 40 700 : 21 750 : 20 40 > 35$

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