How to find out if a number is prime - ISEE Upper Level Quantitative Reasoning

The prime numbers between 10 and 100 that end in a 7 are 17, 37, 47, 67, and 97 (27, 57, and 87 all have 3 as a factor; 77 has 7 as a factor).

Their sum is .

The prime numbers between 50 and 60 are 53 and 59; their sum is . The odd composite numbers between 50 and 60 are 51, 55, and 57; their sum is . Therefore, (B) is greater.

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 greatest prime integer between 30 and 40 is 37, so this is the maximum value of . The greatest prime integer between 20 and 30 is 29, so this is the maximum value of . Since

and ,

then, by the addition property of inequality,

.

We show that it cannot be determined whether is greater than, less than, or equal to 80 by choosing two pairs of primes within the given ranges and adding.

Case 1:

.

Case 2:

In each case, , with and prime.

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

The prime numbers between 80 and 90 are 83 and 89, whose sum is ; the only prime number between 90 and 100 is 97. (A) is the greater quantity.

Out of the ten integers from 41 to 50, the prime numbers are 41, 43, and 47 - therefore, there are three prime numbers and seven composite numbers. Since

,

(B) is greater.

The first two primes after 100 are 101 and 103; their product is

.

The two greatest prime numbers less than 100 are 97 and 89. Their product is:

Every number except 1 has at least two factors - 1 and itself. So we are looking for a number with no other factors - that is, a prime.

3 is a factor of both 33 and 39, and 5 is a factor of 35, so we can eliminate those choices. We cannot eliminate 37, since it is a prime, having only 1 and 37 as factors.

(a) The prime numbers in the to range are , and . Their sum is

.

(b) The prime numbers in the to range are and ; their sum is

.

(a) is greater.

(a) The product of the first five prime numbers is .

(b)

(b) is greater.