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

Which is the greater quantity?

(a) The number of prime numbers between 1 and 20 inclusive

(b) The number of composite numbers between 21 and 30 inclusive

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

(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.

Which is the greater quantity?

(a) The number of prime numbers between 70 and 110

(b) The number of prime numbers between 80 and 120

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

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

Multiply the least and greatest primes between 80 and 100.

Explanation

The least and greatest primes between 80 and 100 are 83 and 97; their product is

$83 \times 97 = 8,051$

Which one is greater?

The number of prime numbers between and

and are equal.

is greater

it is not possible to tell based on the information given.

Explanation

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

The prime numbers between and are:

So and are equal.

Which one is greater?

The number of prime numbers between and

and are equal

is greater

it is not possible to tell based on the information given.

Explanation

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

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.

Which is the greater quantity?

(A) The number of composite integers between 41 and 50 inclusive.

(B) The number of prime integers between 41 and 50 inclusive.

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

The only prime numbers among the ten in the range 41 through 50 are 41, 43, and 47; this makes three prime numbers and seven composite numbers, so (A) is greater.

Add all of the prime numbers between 20 and 40.

Explanation

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

Their sum is .

Which of the following numbers is prime?

Explanation

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

First, find the approximate square root of the number:

$\sqrt{163}\approx12.75\approx13$

We know this because:

$\sqrt{169}=13$

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: .

$\sqrt{147}\approx12$

We know this because:

$\sqrt{144}=12$

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 .

$\frac{147}{3}=49$

, and and are positive integers.

is a prime number; is not a prime number.

Which is the greater quantity?

(a)

(b)

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Explanation

, and and are positive integers, so each of and is an integer from 1 to 11 inclusive.

is a prime number, meaning that it can be equal to 2, 3, 5, 7, or 11. Testing each case:

, which is not prime.

, which is prime - we throw this case out.

, which is not prime.

In the first two cases, ; in the last case, . It cannot be determined which is the greater.

and are prime integers. and . What is the minimum value of ?

Explanation

The least prime integer between 60 and 70 is 61, so this is the minimum value of . The least prime integer between 80 and 90 is 83, so this is the minimum value of . Since