AMC 10 · 2002 · #4

Grade 5 arithmetic

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Integrality / Divisibility Screen

fraction-arithmeticlcmdivisibility-rules bound-inequality-then-enumerate ↑ Prerequisites: fraction-arithmeticlcm

📏 Medium solution 💡 2 insights

Problem

A positive whole number n is chosen so that 1/2+1/3+1/7+1/n comes out to a whole number. Among five statements about n, find the one that is not true.

Pick an answer.

(A)

$\ 2\ \text{divides\ }n$

(B)

$\ 3\ \text{divides\ }n$

(C)

$\ 6\ \text{divides\ }n$

(D)

$\ 7\ \text{divides\ }n$

(E)

$\ n > 84$

How to solve

Strategy Extreme Principle

You cannot judge the five statements until you know n, so the real work is finding n. First (Tool #7, Identify Subproblems) add the three known fractions to get one fixed number, 41/42. Then the key move is Tool #14 (Extreme Principle): because 1/n is small and positive, the running total is squeezed between two limits and can only land on the whole number 1 — that boundary argument forces n=42. With n known, Tool #3 (Eliminate Possibilities) tests each statement and keeps the one that fails.

1STEP 1

Add the three known fractions

Over the common denominator 42 the three known fractions add to 41/42, just under 1.

1/2+1/3+1/7=(21+14+6)/42=41/42

2STEP 2

Trap the total between two limits

Since 1/n is positive and at most 1, the total is trapped strictly between 41/42 and 2, so it must be 1.

41/42 ＜ 41/42+1/n ＜ 41/42+1 ＜ 2 → sum=1

3STEP 3

Solve for n

That leaves 1/n = 1/42, so n = 42.

1/n=1-41/42=1/42 → n=42

4STEP 4

Test each statement against n = 42

Since 42 = 2×3×7 it passes the four divisibility claims, so the false one is n greater than 84.

42=2×3×7 → 2 ∣ 42, 3 ∣ 42, 6 ∣ 42, 7 ∣ 42; 42 ≯ 84 → (E)

Answer

n ＞ 84

Check the found value directly: 1/2+1/3+1/7+1/42=(21+14+6+1)/42=42/42=1, a whole number, so n=42 is correct. Four of the statements (A, B, C, D) are then true and exactly one, (E), is false — matching the question's promise that a single statement fails. The trap answer would be to pick a divisibility statement, but 42 really is a multiple of 2, 3, 6, and 7; only the size comparison n ＞ 84 breaks.

💡Key takeaway

Add the fractions you can, and if what's left is a tiny positive piece, the total can only climb to the next whole number — here that pins n=42, so the false claim is n ＞ 84.

Add the three known fractions
Trap the total between two limits
Solve for n
Test each statement against n = 42

Add the three known fractions

Trap the total between two limits

Solve for n

Test each statement against n = 42

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