AMC 10 · 2003 · #8

Grade 7 probability

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

factorsdivisor-countprobability-basic systematic-enumeration ↑ Prerequisites: factors

📏 Medium solution 💡 2 insights

Problem

You pick one positive factor of 60 at random, with every factor equally likely. Find the probability that the factor you pick is less than 7.

Pick an answer.

(A)

$\frac{1}{10}$

(B)

$\frac{1}{6}$

(C)

$\frac{1}{4}$

(D)

$\frac{1}{3}$

(E)

$\frac{1}{2}$

How to solve

Strategy Make a Systematic List

A probability like this is just a fraction, so the whole job is two counts: how many factors are less than 7 (top) and how many factors there are in all (bottom). Tool #2 (Make a Systematic List) is the safe way to get both without missing or repeating anything — list the factor pairs of 60 in order, and you can read both counts straight off the same list. Tool #7 (Identify Subproblems) keeps the two counts separate so the fraction is assembled cleanly. Tool #16 (Change Focus) gives a fast check: pair each factor d with its partner 60/d, and the pairing shows the small and large factors must split evenly.

1STEP 1

List every factor of 60

Pairing factors that multiply to 60 gives a complete list of 12 factors.

60=1·60=2·30=3·20=4·15=5·12=6·10 → {1,2,3,4,5,6,10,12,15,20,30,60}, 12 factors

2STEP 2

Count the factors below 7

Of those, 1 through 6 are under seven — exactly 6, since 7, 8 and 9 divide nothing here.

{1,2,3,4,5,6} → 6 factors less than 7

3STEP 3

Write the probability as a fraction

Equally likely factors make the probability 6/12.

P=6/12

4STEP 4

Simplify the fraction

Reducing gives 1/2, choice (E).

6/12=1/2 → (E)

Answer

1/2

Exactly half of the 12 factors (6 of them) are less than 7, and half a chance is 1/2 — matching choice (E), and clearly between 0 and 1 as any probability must be. A quick sanity count: the six small factors 1,2,3,4,5,6 sit below 7, and the six large factors 10,12,15,20,30,60 sit above it, a perfect 6-to-6 split.

💡Key takeaway

For a "probability of a factor" question, list every factor in pairs that multiply to the number, count the ones you want and the total, then simplify the fraction.

List every factor of 60
Count the factors below 7
Write the probability as a fraction
Simplify the fraction

List every factor of 60

Count the factors below 7

Write the probability as a fraction

Simplify the fraction

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