If n is the square of a prime number, then it has exactly 3 factors.
Step-by-step solution
To find: Frame the converse of this proposition. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.
Idea: The factors of p2 can only be built from the prime p: 1, p, p2. Going back, 3 factors is an odd number, so n is a square m2, and m must be prime or there would be extra factors.
- Converse: If n has exactly 3 factors, then it is the square of a prime number.½ mark
- Proposition — true. Let n = p2 with p prime. A factor of p × p can contain only the prime p, at most twice, so the factors are 1, p, p2: exactly 3. E.g. 25: 1, 5, 25.1 mark
- Converse — true. 3 is odd, and a number with an odd number of factors is a perfect square. So n = m2, with m > 1 (because 1 has only one factor). Then 1, m and m2 are three different factors, so they are all of them.½ mark
- If m were not prime, it would have a factor d with 1 < d < m; d would be a fourth factor of n. So m is prime and n is the square of a prime. Both statements are true.1 mark
Check: Numbers up to 60 with exactly 3 factors: 4, 9, 25, 49 — exactly the squares of the primes 2, 3, 5, 7. A square of a non-prime, such as 36 = 62, has more (9 factors).
Answer to write in the exam
Converse: If n has exactly 3 factors, then it is the square of a prime number.
Proposition: n = p2 (p prime): factors 1, p, p2 only — exactly 3. True.
Converse: 3 factors (odd) ⇒ n is a perfect square, n = m2, m > 1
1, m, m2 are the 3 factors; if m had a factor d, 1 < d < m, d would be a 4th factor
⇒ m is prime. True.
∴ Both the proposition and its converse are true.
Common mistakes that cost marks
- Listing only 1 and p and forgetting p2 itself.
- Thinking every perfect square has 3 factors. 36 = 62 has 9 factors because 6 is not prime.
- Taking n = 1 = 12 as a counterexample to the proposition. 1 is not a prime number, so 1 is not the square of a prime.
How this can come in the exam
Which number has exactly 3 factors?
- 16
- 25
- 27
- 35
Show answer
(B) 25
25 = 52: factors 1, 5, 25. 16 has 5 factors, and 27 and 35 have 4 each.
Find all the numbers between 1 and 60 that have exactly 3 factors.
Show answer
They are the squares of primes (1 mark): 22 = 4, 32 = 9, 52 = 25, 72 = 49 (112 = 121 is too big) (1 mark).Try one yourself
How many factors does 169 have? List them.
Show answer
169 = 132 and 13 is prime, so it has exactly 3 factors: 1, 13, 169.
More questions like this
- If n is a product of two unequal prime numbers, then it has exactly 4 divisors.
- If n and n + 3 have no factors in common, then n is not a multiple of 3.
- There are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims.
- Find counterexamples to the following statements.
- Consider the statement: ‘If a number is divisible by 8, then it is divisible by both 2 and 4’.