A book distributor is trying to divide an order of textbooks into equally sized groups for shipping in cartons. The textbooks can be divided into groups of 12, groups of 15, or groups of 18, with no books left over. Which of the following inequalities is satisfied if \( N \) is the smallest possible total number of textbooks?
A. \( 100 < N < 150 \)
B. \( 150 < N < 200 \)
C. \( 200 < N < 250 \)
D. \( 250 < N < 300 \)
Answer (3)
calculate least common factor of 12, 15 & 18. It will be 180. answer is B. 150 < N < 200
The question concerns finding the smallest number N of textbooks that can be evenly divided into groups of 12, 15, or 18 with no leftovers. The smallest such N should be the least common multiple (LCM) of 12, 15, and 18. To find this, we look at the prime factors of each number:
12 = 2² × 3
15 = 3 × 5
18 = 2 × 3²
The LCM is therefore 2² × 3² × 5 = 4 × 9 × 5 = 180. Since 180 can be divided evenly by 12, 15, and 18, it is our smallest possible total for the number of textbooks. Therefore, option C, 200 < N < 250, cannot be correct as N must be 180 to satisfy the conditions, and this falls within the range of option B, 150 < N < 200. Thus, the inequality that is satisfied for the smallest possible total number of textbooks would be represented by option B.
The smallest number of textbooks that can be divided into groups of 12, 15, or 18 with no leftovers is 180. This value falls within the range specified in option B: 150 < N < 200. Thus, the answer is option B.
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