AMC 10 · 2003 · #8
Grade 7 probabilityWhat is the probability that a randomly drawn positive factor of 60 is less than 7?
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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 $7$.
Givens: The number is $60$; A positive factor of $60$ is a whole number that divides $60$ with no remainder; Every positive factor of $60$ is equally likely to be drawn; Answer choices: (A) $\frac{1}{10}$, (B) $\frac{1}{6}$, (C) $\frac{1}{4}$, (D) $\frac{1}{3}$, (E) $\frac{1}{2}$
Unknowns: The probability that the drawn factor is less than $7$
Understand
Restated: 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 $7$.
Givens: The number is $60$; A positive factor of $60$ is a whole number that divides $60$ with no remainder; Every positive factor of $60$ is equally likely to be drawn; Answer choices: (A) $\frac{1}{10}$, (B) $\frac{1}{6}$, (C) $\frac{1}{4}$, (D) $\frac{1}{3}$, (E) $\frac{1}{2}$
Plan
Primary tool: #2 Make a Systematic List
Secondary: #7 Identify Subproblems, #16 Change Focus / Count the Complement
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.
Execute — Answer: E
4.OA.B.4 Step 1 List every factor of 60
- Build the factors in pairs that multiply to $60$: $1\times 60$, $2\times 30$, $3\times 20$, $4\times 15$, $5\times 12$, $6\times 10$.
- After $6\times 10$ the two numbers in each pair would cross over, so the list is complete.
- Writing them in order gives $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$.
- Counting them, $60$ has $12$ factors in all.
💡 Listing factors as pairs that multiply to $60$ guarantees you catch all of them and stop at the right place.
4.OA.B.4 Step 2 Count the factors below 7
- From the same list, pick out the factors that are less than $7$ (that is, $6$ or smaller): $1, 2, 3, 4, 5, 6$.
- There are $6$ of them.
- Notice the next factor after $6$ jumps straight to $10$ — the numbers $7, 8, 9$ are not factors of $60$, so nothing between $6$ and $10$ is missed.
💡 The count you want is just the front part of the ordered list, up to the last factor below $7$.
7.SP.C.7 Step 3 Write the probability as a fraction
- Because every factor is equally likely, the probability is the number of factors you want divided by the total number of factors.
- That is $\dfrac{\text{factors less than }7}{\text{all factors}}=\dfrac{6}{12}$.
💡 With equally likely outcomes, probability is simply the favorable count over the total count.
4.NF.A.1 Step 4 Simplify the fraction
- Reduce $\dfrac{6}{12}$ by dividing top and bottom by $6$: $\dfrac{6}{12}=\dfrac{1}{2}$.
- So the probability that a randomly drawn factor of $60$ is less than $7$ is $\dfrac{1}{2}$, which is choice (E).
💡 Dividing numerator and denominator by the same number keeps the fraction's value while showing its simplest form.
4.OA.B.4 Build the factors in pairs that multiply to $60$: $1\times 60$, $2\times 30$, $3 4.OA.B.4 From the same list, pick out the factors that are less than $7$ (that is, $6$ or 7.SP.C.7 Because every factor is equally likely, the probability is the number of factors 4.NF.A.1 Reduce $\dfrac{6}{12}$ by dividing top and bottom by $6$: $\dfrac{6}{12}=\dfrac{ Review
Reasonableness: Exactly half of the $12$ factors ($6$ of them) are less than $7$, and half a chance is $\frac{1}{2}$ — 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.
Alternative: Use the pairing idea (tool #16). Every factor $d$ of $60$ pairs with the factor $60/d$, and the two are never equal because $60$ is not a perfect square. In each pair one member is at most $6$ and the other at least $10$ (for example $6\leftrightarrow10$, $5\leftrightarrow12$, $1\leftrightarrow60$). So the factors split into exactly as many below $7$ as above $7$, which forces the probability to be exactly $\frac{1}{2}$ — the same answer (E) — without even finishing the full list.
CCSS standards used (min grade 7)
4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing all $12$ factors of $60$ as factor pairs and picking out the $6$ that are less than $7$.)7.SP.C.7Develop probability models and use them to find probabilities of events (Writing the probability as favorable factors over total factors, $\frac{6}{12}$, under equally likely outcomes.)4.NF.A.1Explain why a fraction is equivalent to another fraction (Reducing $\frac{6}{12}$ to its simplest form $\frac{1}{2}$.)
⭐ 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.
⭐ 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.
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