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Difficulty:
75%
(hard)
Question Stats:
52% (02:35) correct
48%
(02:29)
wrong
based on 98
sessions
History
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Time
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Determine the smallest 5-digit number that, when divided by 72, 84, and 48 respectively, leaves remainders of 53, 65, and 29.
A. 10,016
B. 10,061
C. 10,601
D. 10,610
E. 16,001
(Source=TCYOnline)
A. 10,016
B. 10,061
C. 10,601
D. 10,610
E. 16,001
(Source=TCYOnline)
Right Answer Explanation / Suggested Answer :Let N be the given number.
N leaves a remainder of 53 when divided by 72, i.e., for
N/72, the remainder is 53.
So, N + 19 will be divisible by 72.
Further, N leaves a remainder of 65 when divided by 84, i.e., for
N/84, the remainder is 65. Therefore, N + 19 will be divisible by 84.
N leaves a remainder of 29 when divided by 48, i.e., for
N/48, the remainder is 29. Hence, N + 19 will be divisible by 48.
Thus, N + 19 will be divisible by 72, 84, and 48, i.e., N + 19 is a multiple of 72, 84, and 48.
The least value of (N + 19) is the LCM of 72, 84, and 48, which is 1,008.
Since this is a 4-digit number, to find the least 5-digit number, multiply the LCM by 10 to get 10,080.
Subtracting 19, we get 10,080 - 19 = 10,061 as the required number.
N leaves a remainder of 53 when divided by 72, i.e., for
N/72, the remainder is 53.
So, N + 19 will be divisible by 72.
Further, N leaves a remainder of 65 when divided by 84, i.e., for
N/84, the remainder is 65. Therefore, N + 19 will be divisible by 84.
N leaves a remainder of 29 when divided by 48, i.e., for
N/48, the remainder is 29. Hence, N + 19 will be divisible by 48.
Thus, N + 19 will be divisible by 72, 84, and 48, i.e., N + 19 is a multiple of 72, 84, and 48.
The least value of (N + 19) is the LCM of 72, 84, and 48, which is 1,008.
Since this is a 4-digit number, to find the least 5-digit number, multiply the LCM by 10 to get 10,080.
Subtracting 19, we get 10,080 - 19 = 10,061 as the required number.
Amity007
Nice one — this tests LCM + remainder reasoning, and the trick is spotting what the three remainder conditions have in common.
Step 1: Spot the pattern in the remainders.
N ÷ 72 leaves remainder 53
N ÷ 84 leaves remainder 65
N ÷ 48 leaves remainder 29
Notice: 72 − 53 = 19, 84 − 65 = 19, 48 − 29 = 19.
Each time, N is exactly 19 less than a full multiple of the divisor. That means N + 19 is divisible by all three: 72, 84, and 48.
Step 2: Find LCM(72, 84, 48).
Prime factorizations:
- 72 = 23 × 32
- 84 = 22 × 3 × 7
- 48 = 24 × 3
LCM = 24 × 32 × 7 = 16 × 9 × 7 = 1,008
Step 3: Set up the equation.
N + 19 = 1,008k (for some positive integer k)
N = 1,008k − 19
Step 4: Find the smallest 5-digit N.
We need N ≥ 10,000:
1,008k − 19 ≥ 10,000
1,008k ≥ 10,019
k ≥ 9.94...
So k = 10 → N = 1,008(10) − 19 = 10,080 − 19 = 10,061
Step 5: Quick verify.
10,061 ÷ 72 = 139 remainder 53 ✓
10,061 ÷ 84 = 119 remainder 65 ✓
10,061 ÷ 48 = 209 remainder 29 ✓
Common trap: Students who don't spot the "constant deficit" pattern end up trying to work with three separate remainder conditions — which is messy and slow. The moment you see that the gap between the divisor and remainder is the same across all conditions, you know to add that constant and find the LCM. This is a huge time-saver.
Takeaway: When a problem gives you multiple divisor-remainder pairs, always check (divisor − remainder) first. If that difference is constant, the problem collapses into a single LCM calculation.
Answer: B (10,061)
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Step 1: Spot the pattern in the remainders.
N ÷ 72 leaves remainder 53
N ÷ 84 leaves remainder 65
N ÷ 48 leaves remainder 29
Notice: 72 − 53 = 19, 84 − 65 = 19, 48 − 29 = 19.
Each time, N is exactly 19 less than a full multiple of the divisor. That means N + 19 is divisible by all three: 72, 84, and 48.
Step 2: Find LCM(72, 84, 48).
Prime factorizations:
- 72 = 23 × 32
- 84 = 22 × 3 × 7
- 48 = 24 × 3
LCM = 24 × 32 × 7 = 16 × 9 × 7 = 1,008
Step 3: Set up the equation.
N + 19 = 1,008k (for some positive integer k)
N = 1,008k − 19
Step 4: Find the smallest 5-digit N.
We need N ≥ 10,000:
1,008k − 19 ≥ 10,000
1,008k ≥ 10,019
k ≥ 9.94...
So k = 10 → N = 1,008(10) − 19 = 10,080 − 19 = 10,061
Step 5: Quick verify.
10,061 ÷ 72 = 139 remainder 53 ✓
10,061 ÷ 84 = 119 remainder 65 ✓
10,061 ÷ 48 = 209 remainder 29 ✓
Common trap: Students who don't spot the "constant deficit" pattern end up trying to work with three separate remainder conditions — which is messy and slow. The moment you see that the gap between the divisor and remainder is the same across all conditions, you know to add that constant and find the LCM. This is a huge time-saver.
Takeaway: When a problem gives you multiple divisor-remainder pairs, always check (divisor − remainder) first. If that difference is constant, the problem collapses into a single LCM calculation.
Answer: B (10,061)
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