What is the sum of all numbers less than 200 that are either prime or

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Recall that the number 1 has only 1 factor and all the prime numbers have only 2 factors. All the other positive integers (i.e., composite numbers) have more than 2 factors. Therefore, in order to find the sum of all the numbers less than 200 that are either prime or have more than 3 factors, we must exclude 1 and all the numbers that have exactly 3 factors. The numbers that have exactly 3 factors are those that are squares of a prime (for example, 2^2 = 4 has exactly 3 factors: 1, 2 and 4). The sum of excluded numbers is:

1 + 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 1 + 4 + 9 + 25 + 49 + 121 + 169 = 378

The sum of all positive integers less than 200 is:

1 + 2 + 3 + … + 198 + 199 = 199(199 + 1)/2 = 199(200)/2 = 199(100) = 19,900

Therefore, the sum of all the numbers less than 200 that are either prime or have more than 3 factors is 19,900 - 378 = 19,522.

Answer: E

What is the sum of all numbers less than 200 that are either prime or have more than 3 factors?

If we take out 1 and square of prime nos. then left numbers below 200 will be either prime or having more factors than 3.

Sum of all nos. below 200 = 199*200/2 = 19900
Sum to be excluded = 1+ 4+9+25+49 + 121 +169 = 378

Required sum = 19900- 378 = 19522. So, I think E.

Firstly, the sum of the first 199 natural numbers: 199*200/2=19900.
We have to exclude numbers which are either prime or have less than 3 factors. Such numbers are 1 and the square of prime numbers 2,3,5,7,11,13.
So, 19900-(1+4+9+25+49+121+169)=19522(E).