Prime Numbers - ISEE Upper Level: Quantitative Reasoning

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.

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.

(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.

The two greatest primes less than or equal to 100 are 97 and 89; their product is , making (b) greater.

The two least primes greater than or equal to 50 are 53 and 59; their product is .

The five greatest primes less than or equal to 100 are . Their sum is

A prime number is a number that is only divisible by itself and .

is the only number that is not prime.

A prime number is a natural number greater than that can be divided only by and itself. The smallest prime number greater than is and the smallest prime number greater than is . So we can write:

So is greater than .

A prime number is a natural number greater than that can be divided only by and itself.The prime numbers between and are:

That means there are four prime numbers between and . So and are equal.

A prime number is a natural number greater than that can be divided only by and itself.

The prime numbers between and are:

The prime numbers between and are:

So and are equal.

A prime number is a natural number greater than that can be divided only by and itself.

The prime numbers between and are:

The prime numbers between and are:

So there are five prime numbers between and and there are five prime numbers between and too. Therefore and are equal.

A prime number is a natural number greater than that can be divided only by and itself. The first four prime numbers are: .

So their product is:

, which is greater than .