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06 Feb 2026, 15:53
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
58% (02:58) correct
42%
(02:44)
wrong
based on 48
sessions
History
Date
Time
Result
Not Attempted Yet
Three friends Ryan, Jason, and Sharon met with a car accident in November 2019 and were admitted to a hospital. They were discharged from the hospital on the 1st day of 2020. Depending on the pace of recovery, the doctor advised them to visit the hospital for a complete year. Ryan needed to visit every third day, Jason every fourth day, and Sharon every fifth day. How many days of the current year would exactly two of the friends visit the hospital?
A. 24 days
B. 45 days
C. 50 days
D. 54 days
E. 72 days
(Source=TCYOnline)
A. 24 days
B. 45 days
C. 50 days
D. 54 days
E. 72 days
(Source=TCYOnline)
06 Feb 2026, 23:30
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This is a classic inclusion-exclusion problem on LCMs — and the key phrase to watch is "exactly two."
Step 1: Identify the visit cycles.
Starting Jan 1, 2020 (day 0), over a full year (2020 is a leap year, so 366 days):
- Ryan visits every 3rd day: days 3, 6, 9, ...
- Jason visits every 4th day: days 4, 8, 12, ...
- Sharon visits every 5th day: days 5, 10, 15, ...
Step 2: Count individual visits.
- Ryan: ⌊366/3⌋ = 122 days
- Jason: ⌊366/4⌋ = 91 days
- Sharon: ⌊366/5⌋ = 73 days
Step 3: Count pairwise overlaps (days when at least two visit together).
- Ryan AND Jason: LCM(3,4) = 12 → ⌊366/12⌋ = 30 days
- Ryan AND Sharon: LCM(3,5) = 15 → ⌊366/15⌋ = 24 days
- Jason AND Sharon: LCM(4,5) = 20 → ⌊366/20⌋ = 18 days
Step 4: Count days when ALL three visit.
LCM(3, 4, 5) = 60 → ⌊366/60⌋ = 6 days
Step 5: Apply the "exactly two" formula.
Each day when all three meet is counted once in each of the three pairwise counts. We want to remove those days entirely from our "exactly two" count:
Exactly two = (30 + 24 + 18) − 3 × 6 = 72 − 18 = 54 days
Common trap: The biggest mistake is computing "at least two" (which is 72) instead of "exactly two" (54). When you see "exactly" in a counting problem, you must always subtract the triple overlap. Students who skip Step 5 land on answer choice E (72) — which is there on purpose as a trap answer.
Takeaway: For any "exactly k out of n" overlap problem, always start with inclusion-exclusion and be precise about stripping out higher-order overlaps. Writing out the Venn diagram for 30 seconds can save you from a careless error.
Answer: D (54 days)
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Free gamified GMAT prep: edskore.com | 725 GMAT Focus
Step 1: Identify the visit cycles.
Starting Jan 1, 2020 (day 0), over a full year (2020 is a leap year, so 366 days):
- Ryan visits every 3rd day: days 3, 6, 9, ...
- Jason visits every 4th day: days 4, 8, 12, ...
- Sharon visits every 5th day: days 5, 10, 15, ...
Step 2: Count individual visits.
- Ryan: ⌊366/3⌋ = 122 days
- Jason: ⌊366/4⌋ = 91 days
- Sharon: ⌊366/5⌋ = 73 days
Step 3: Count pairwise overlaps (days when at least two visit together).
- Ryan AND Jason: LCM(3,4) = 12 → ⌊366/12⌋ = 30 days
- Ryan AND Sharon: LCM(3,5) = 15 → ⌊366/15⌋ = 24 days
- Jason AND Sharon: LCM(4,5) = 20 → ⌊366/20⌋ = 18 days
Step 4: Count days when ALL three visit.
LCM(3, 4, 5) = 60 → ⌊366/60⌋ = 6 days
Step 5: Apply the "exactly two" formula.
Each day when all three meet is counted once in each of the three pairwise counts. We want to remove those days entirely from our "exactly two" count:
Exactly two = (30 + 24 + 18) − 3 × 6 = 72 − 18 = 54 days
Common trap: The biggest mistake is computing "at least two" (which is 72) instead of "exactly two" (54). When you see "exactly" in a counting problem, you must always subtract the triple overlap. Students who skip Step 5 land on answer choice E (72) — which is there on purpose as a trap answer.
Takeaway: For any "exactly k out of n" overlap problem, always start with inclusion-exclusion and be precise about stripping out higher-order overlaps. Writing out the Venn diagram for 30 seconds can save you from a careless error.
Answer: D (54 days)
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Free gamified GMAT prep: edskore.com | 725 GMAT Focus
