Understanding the Smallest 3-Digit Number Divisible by 3 and Another Factor
When tackling the problem of finding the smallest 3-digit number divisible by 3 and another factor, it’s essential to consider the mathematical properties and the Least Common Multiple (LCM). This guide will walk you through various strategies and examples to help you find the solution accurately.
Example 1: Divisibility by 3, 72, and 743
One of the examples provided includes the question of finding the smallest 3-digit number divisible by 3 and 743. However, 743 and 742 (148) are not divisible by 3. The next number, 741, is 3×247, so it’s not divisible by 3. Therefore, the number 743 is not divisible by 3 and 742 (148). Hence, the answer ends up being 222. Yet, this approach seems inaccurate since it does not converge on a correct multiple of both 3 and 743.
Example 2: Divisibility by 3, 72, and 144
Another example that looks for the first three-digit number in the table of 72, which is 144. Since 144 is divisible by 3 (144 ÷ 3 48), the smallest 3-digit number divisible by both 3 and 72 is 144.
Example 3: Divisibility by 3 and 77
For finding the LCM of 3 and 77, the least common multiple (LCM) is calculated as 3×7×11, which is 231. Therefore, the smallest 3-digit number divisible by both 3 and 77 is 231.
Example 4: Divisibility by 3 and 75
When we consider the problem of finding the smallest 3-digit number divisible by 3 and 75, we first note that since 3 divides 75, we need to find the smallest 3-digit number divisible by 75. Dividing 1000 by 75 gives approximately 13.3, and 13 times 75 equals 975, making it the smallest three-digit number divisible by both 3 and 75.
Example 5: Divisibility by 3, 75, and 975
Since 3 divides 75, we need only find the smallest three-digit number divisible by 75. The smallest 3-digit number divisible by 75 is 975, which is 13 times 75. Thus, 975 is the smallest 3-digit number divisible by 3 and 75.
Example 6: Divisibility by 3, 75, and 150
Dividing 100 by 75 gives 1.33, and 13 times 75 equals 1050, which is not a 3-digit number. The next multiple of 75 that is a 3-digit number is 975. Simplifying, 75 ÷ 3 25, and 100 ÷ 50 2, making the smallest 3-digit number divisible by 3 and 75 equal to 150.
Conclusion
The process of finding the smallest 3-digit number divisible by 3 and another factor involves identifying the least common multiple (LCM) and finding the smallest 3-digit multiple of that LCM. Whether it’s 144, 231, or 150, understanding the LCM and divisibility rules is vital for solving such problems accurately.