AMC Rules of Prime Numbers Part - 2

Which of the following is NOT an element of both the set of of positive odd integers and set of prime numbers?

• 7

• 17

• 37

• 47

• 57

How many integers greater than 20 and less than 30 are each the product of exactly two prime numbers?

• 3

• 4

• 5

• 6

• 7

The sum of three consecutive prime numbers is 159. What is the largest prime number?

• 53

• 57

• 59

• 61

• 47

Answer explanation

The three consecutive prime numbers that sum to 159 are 53, 59, and 61. Among these, the largest prime number is 59.

How many pairs of distinct prime numbers have a sum of 22?

• 1

• 2

• 3

• 4

• 5

Answer explanation

The distinct prime pairs that sum to 22 are (3, 19) and (5, 17). Thus, there are 2 pairs of distinct prime numbers that satisfy the condition.

How many prime numbers satisfy √300 < x < √700?

• 0

• 1

• 2

• 3

• 4

Answer explanation

To find prime numbers between √300 (approximately 17.32) and √700 (approximately 26.46), we check the integers 18 to 26. The prime numbers in this range are 19 and 23, giving us a total of 2 prime numbers.

An ordered pair refers to a pair of two numbers (or variables) written inside brackets and are separated by a comma. For example, (1, 2) is an ordered pair.

How many ordered pairs of primes (a, b) satisfy the equation a + b = 20?

• 1

• 4

• 3

• 2

• 5

Answer explanation

The prime pairs that satisfy a + b = 20 are (3, 17), (7, 13), (11, 9), and (9, 11). However, only (7, 13) and (13, 7) are valid since 9 is not prime. Thus, there are 4 valid ordered pairs of primes, e.g. (3, 17), (17, 3), (7,13), (13, 7).

For how many ordered triplets of primes (a, b, c); a+b+c = 26; where a ≤ b ≤ c.

• 1

• 2

• 3

• 4

• 5

Answer explanation

To find ordered triplets of primes (a, b, c) such that a + b + c = 26 and a ≤ b ≤ c, we can see that the sum of three prime numbers is an even number. So, one of them must be 2. Since 'a' is the smallest prime, therefore, a = 2 and b+c = 24. The valid triplets are (2, 3, 11), (2, 5, 19), and (2, 7, 17).

If a + b + C = 28, a< b<c, and ab + bc + ca = 185, where a, b, c all are prime numbers, find the greatest of the three prime numbers. (Hint: The sum of the three prime numbers is an even number)

• 21

• 12

• 17

• 19

• 26

Answer explanation

According to the rule, if the sum of three prime numbers is an even number then one of them must be 2.

Since "a" is the smallest so, a = 2.

Given a + b + c = 28, (b+c) = 26 and ab + bc + ca = 185, OR 2b + bc + 2c = 185, or bc + 2 (b + c) = 185.

So, bc = 133 = 7 x 19

The sum of the square of a prime number and an odd positive integer is 125. What is that odd positive integer?

• 121

• 131

• 141

• 151

• 161

If p, q, and r all are prime numbers, p and q are consecutive prime numbers where p < q and p² + q² = r, then find the value of q.

• 17

• 19

• 2

• 3

• 5