Prime Numbers - ISEE Upper Level Quantitative Reasoning

The prime numbers between 20 and 40 are 23, 29, 31, and 37.

Their sum is .

The prime numbers between 50 and 70 are 53, 59, 61, and 67. Their sum is

.

A prime number has exactly two factors, 1 and itself. We can eliminate four choices by finding other factors:

53 has only 1 and 53 as factors, so it is the only prime among the choices.

There are twenty integers from to inclusive. Counting composite numbers can be made easier by weeding out the primes - . Removal of these five primes leaves composite numbers.

The correct answer is , and this can be determined in the following manner.

First, find the approximate square root of the number:

We know this because:

Therefore, we only need to consider prime numbers through

Is evenly divisible by any of these numbers? In this case, the answer is no, therefore is prime. Consider the case where the answer is not prime: .

We know this because:

Therefore, we need to consider the followig prime numbers:

Is divisible by any of these numbers? In this case, the answer is yes. is divisible by .

There are twenty integers from 61 to 80 inclusive. Counting composite numbers can be made easier by weeding out the primes - 61, 67, 71, 73, 79. Removal of these five primes leave fifteen composite numbers.

There are thirty integers from to . Counting composite numbers can be made easier by weeding out the primes - . There are six primes in this range, so this leaves twenty-four composite numbers.

The correct answer to this question is . To determine if a number is prime, first find the approximate square root of the number:

Next, determine if the number is divisible by any prime numbers up to and including the square root. In this instance consider the numbers:

is not divisible by any of the above numbers; therefore, it is prime.

First, given that 13 times 4 is 52, the numerator can be simplified by taking the square root of 4 and multiplying it by the square root of 13.

A prime number is one in which only itself and 1 can go into it. Therefore, 2, 3, 5, 7, 11, 13, 17, and 19 are the only prime numbers in that set.

Which of the following is a prime number?

The answer here is 13, because 13 has no factors other than 1 and itself.

51 is divisible by 3 and 17.

39 is divisible by 13 and 3.

81 is divisible by 9, 3, and 27

The prime numbers between and are , , , and . Their sum is:

.

The primes between 20 and 40 are included in both sets, so all we need to do is to compare the number of primes between 40 and 50 with the number of primes between 10 and 20.

(a) The primes between 40 and 50 are 41, 43, and 47 - three.

(b) The primes between 10 and 20 are 11, 13, 17, and 19 - four.

Since the number of primes between 10 and 20 outnumbers those between 40 and 50, the number of primes between 10 and 40 outnumber those between 20 and 50. Therefore, (b) is the greater.

The primes between 80 and 110 are included in both sets, so all we need to do is to compare the number of primes between 70 and 80 and the number of primes between 110 and 120.

(a) The primes between 70 and 80 are 71, 73, and 79 - three primes

(b) The only prime between 110 and 120 is 113.

(a) is the greater quantity

The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total. Since 1 is neither prime nor composite, this leaves 11 composite numbers. (b) is the greater quantity.

(a) The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total.

(b) The prime numbers between 21 and 30 inclusive are 23 and 29 - two prime numbers out of ten integers. This leaves eight composite numbers.

(a) and (b) are therefore equal.

(a) The prime numbers in the 11-19 range are 11, 13, 17, and 19. Their sum:

(b) The composite numbers in the 11-19 range are 12, 14, 15, 16, and 18. Their sum:

(b) is greater.

It is not necessary to count all of the composite numbers in the 1-1,000 range. Except for 2, every even number is composite, and there are more than one odd composite numbers (15, 25) to make up for 2. This makes (a) greater.

The prime integers between 75 and 100 are

Their sum is

The even integers from 21 to 30 inclusive are 22, 24, 26, 28, and 30; their sum is

The prime integers from 21 to 30 inclusive are 23 and 29. their sum is

and three times this sum is

This makes (B) greater