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What is the smallest 4-digit number which, when divided by 12, 16, and 20, gives a remainder of 9?
1209
1217
1017
1049
1209
Find the LCM of the divisors (12, 16, 20) and add the common remainder (9) to it. Then, find the smallest multiple of this LCM that result in a 4-digit number.
Find the LCM of the divisors (12, 16, 20) and add the common remainder (9) to it. Then, find the smallest multiple of this LCM that result in a 4-digit number.
Calculate LCM
Find the Least Common Multiple of 12, 16, and 20. 12=223, 16=2222, 20=22*5. LCM = 2⁴ * 3 * 5 = 16 * 15 = 240.
Find 4-Digit Multiple
Multiples of 240 are 240, 480, 720, 960, 1200. The smallest 4-digit multiple of 240 is 1200.
Add Remainder
Since the remainder is 9, we add 9 to the smallest 4-digit multiple of the LCM: 1200 + 9 = 1209.
B: 1217 leaves remainder 17 instead of 9 when divided by 12, 16, 20. C: 1017 is not a multiple of 240 + 9 (1017 - 9 = 1008, 1008/240 is not an integer). D: 1049 leaves a remainder of 49 instead of 9.
A is correct because 1209 divided by 12, 16, and 20 consistently leaves a remainder of 9, and it is the smallest 4-digit number satisfying this condition.
Always verify the remainder condition by dividing the result by the smallest divisor (e.g., 12) first to quickly eliminate wrong choices.
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