"A Necklace Is Made by Stringing N Individual Beads..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

A necklace is made by stringing $N$ individual beads together in the repeating pattern red bead, green bead, white bead, blue bead, and yellow bead. If the necklace design begins with a red bead and ends with a white bead, then $N$ could equal

(A) 16

(B) 32

(C) 41

(D) 54

(E) 68

Try it before reading on.

Step 1: Identify the Repeating Cycle

The pattern is: red, green, white, blue, yellow — then it repeats.

The cycle length is 5. Every 5 beads, the pattern starts over with red.

When a sequence problem has answer choices above 20, there's almost always a pattern that makes brute-force counting unnecessary. Writing out 68 beads would take too long and introduce counting errors. The pattern is the shortcut.

Step 2: Determine Which Positions End on White

The necklace starts with red (position 1) and ends on white. Looking at the cycle, white appears at positions 3, 8, 13, 18, 23, 28...

The pattern: start at 3, then add 5 each time.

Step 3: Match Against the Answer Choices

Now check which answer choices appear in that list:

(A) 16 — not in the list

(B) 32 — not in the list

(C) 41 — not in the list

(D) 54 — not in the list

(E) 68 — in the list

The answer is (E).

Alternative Approach: Remainders

There's a second way to see this. The cycle repeats every 5 beads. White is the 3rd bead in the cycle. So a necklace ending on white means the total number of beads, $N$, leaves a remainder of 3 when divided by 5.

Check each answer:

Only 68 leaves a remainder of 3. The answer is (E).

Both methods — the last-digit pattern and the remainder approach — come from the same insight: the sequence repeats every 5 beads, and white is 3 positions into the cycle. Use whichever method clicks faster for you.

Why This Problem Matters

About 20% of test takers miss this problem. The errors aren't primarily math errors — they're approach errors. People who miss this one tend to fall into one of these traps:

Brute-force attempts: Trying to write out all 68 beads to see what color the last one is. This takes too long, and by the time you reach bead 40, you've probably miscounted at least once. When answer choices are above 20, switch to pattern-finding.

Wrong cycle length: If you misidentify the cycle — say, thinking it repeats every 4 instead of every 5 — every subsequent calculation is likely to be wrong. The cycle length is the number of distinct items before the pattern restarts. Here, that's 5: red, green, white, blue, yellow.

Off-by-one on the position: White is position 3 in the cycle, not position 2 or 4. If you count red as position 0 instead of position 1, the remainder you're looking for shifts. Red is position 1, so white is position 3 — remainder 3 when divided by 5.

The pattern-finding system works because the GMAT® designs sequence problems to be solvable in about two minutes. If your approach would take five minutes, there's a pattern you haven't found yet. The cycle length is the first thing to identify — everything else follows from that.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Percent, Integer, and Sequence Problems

From Episode 24 of Real GMAT® Problems (The GMAT® Strategy Podcast).