1
Smallest prime and smallest composite
Not answeredWhat is the smallest prime number? What is the smallest composite number?
Lesson 17.7 — Chapter 17 Exercise
A complete English student edition of Test 17. Work through prime definitions, parity, prime equations, digit constructions, reciprocal sums, and extended prime searches.
Exercise directions
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Tips for this exercise
Prime and composite conventionsA prime is a whole number greater than 1 and has exactly two positive divisors. A composite is a whole number greater than 1 and has more than two. The numbers 0 and 1 are neither prime nor composite.
Show complete reasoningQuestions 11–13 are extended-response problems. Enter the requested values and explain why your candidates work and why earlier or alternative candidates fail.
Section I · Fill-in questions
Questions 1–10 · 6 points each · 60 points
2
Three distinct primes total 14
Not answeredThree different prime numbers have a sum of 14. What is their product?
product
Optional prime list and calculation
3
A three-prime arithmetic pattern
Not answeredIf p, p+20, and p+40 are all prime, find p.
prime
Optional remainder reasoning
4
Smallest composite for every ones digit
Not answeredFor each ones digit from 0 through 9, write the smallest composite number that ends in that digit. One-digit composite numbers are allowed; for example, 4 itself ends in 4. The worked example gives this example: the smallest composite ending in 5 is 15.
Optional search notes
5
Product equals seven times the sum
Not answeredThe product of three prime numbers is exactly seven times their sum. Find the three primes.
Enter the primes in increasing order.
Optional equation or factor reasoning
6
Weighted sum of two primes
Not answeredThree times one prime plus twice another prime equals 12. What is the product of the two primes?
3p + 2q = 12
p × q
Optional parity reasoning
7
Greatest three-digit prime from a digit pool
Not answeredChoose three different digits from the pool below and arrange them to form the greatest possible three-digit prime.
123458
three digits
Optional candidate search
8
Prime neighbors six away
Not answeredCount the primes p < 50 for which both p−6 and p+6 are also prime.
The limit of 50 applies to p only. The number p+6 may be greater than 50.
primes
Optional list of qualifying primes
9
Recover primes from a reciprocal sum
Not answeredThe sum of the reciprocals of three prime numbers is
1661 / 1986
What is the sum of the three primes?
sum
Optional denominator factorization
10
Three primes with x+y=z
Not answeredThree prime numbers x, y, z satisfy:
x + y = z
What is the smallest of the three primes?
prime
Optional parity explanation
Section II · Extended-response questions
Questions 11–13 · 20 points each · 60 points
11
Concatenate two primes
Not answeredWrite a one-digit prime a to the right of a two-digit prime b, forming a three-digit number. The three-digit number is 87 times a. Find a and b.
Optional hint: writing one number after another
For example, writing 6 to the right of 24 makes 246: multiply 24 by 10, then add 6.
Use the same place-value idea here:
three-digit number = 10b + a = 87a
12
When a squared-number expression is prime
Not answeredLet a be a natural number (a whole number 0, 1, 2, …). If
(a × a − 4) ÷ 7
is a prime number, what are the two smallest possible values of a?
Enter the two values in increasing order.
13
Five primes separated by 6
Not answeredGive five prime numbers in increasing order so that the difference between every two adjacent numbers is 6.