Number 3
4 marks · HardA number n is such that it has exactly 4 positive factors, and n is less than 100.
List all possible values of n and explain your reasoning.
List all the numbers less than 100 with exactly 4 positive factors.
Explain how you determined these numbers.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a) The numbers are 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 93, 94, 95.
(b) Any number with exactly four positive factors must be either the product of two distinct prime numbers (p × q), or a cube of a prime number (p^3).
Also accept: 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 93, 94, 95, 97
Common mistakes: Including numbers with more than 4 factors, or missing some
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Marking points
M1 (a) · 1 mark · Shows correct process for finding numbers with exactly 4 factors.
A1 (a) · 1 mark · Correct list of all such numbers less than 100.
M2 (b) · 1 mark · Explains that a number with 4 factors is of the form p×q (distinct primes) or p^3.
A2 (b) · 1 mark · Links reasoning to factor count for two forms: (1, p, q, pq) and (1, p, p^2, p^3).
Worked solution
A number with four factors is either the product of two distinct primes (pq), or a cube of a prime (p^3).
Primes less than 100: 2, 3, 5, 7, 11, ...
p × q: e.g., 2×3=6, 2×5=10, 2×7=14, 2×11=22, 3×5=15, 3×7=21, 3×11=33, ... List all such products less than 100.
p^3: 2^3=8, 3^3=27, 5^3=125 (>100), so only 8 and 27.
Thus, numbers are 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 93, 94, 95 (pq), plus 8, 27 (p^3).
Reasoning: pq (distinct primes) has factors 1, p, q, pq. p^3 has 1, p, p^2, p^3.