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Details and assumptions
The number is a 2 digit number, not a 3 digit number.
KEY TECHNIQUES
Least Common Multiple (LCM), Chinese Remainder Theorem
ANSWER
122
What is the smallest positive 3 digit number that leaves a remainder of 2 when divided by 3, 4, 5 or 6?
12 = 012
Let be the 3 digit number. We are given that is a multiple of 3, 4, 5 and 6. Hence, must be a multiple of the Lowest Common
Multiple, which is 60. This implies or for some integer .
Let's check for various values of .
, is not a 3 digit number.
, is not a 3 digit number.
, is a 3 digit number.
Thus, 122 is the smallest 3 digit positive integer which satisfies the conditions.
N N − 2 N − 2
N − 2 = 60k N = 60k + 2 k
k
k = 0 N = 60 × 0 + 2 = 2
k = 1 N = 60 × 1 + 2 = 62
k = 2 N = 60 × 2 + 2 = 122
2 Left behind
40 points
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6/8/2013https://brilliant.org/assessment/s/algebra-and-number-theory/2300621/