Common multiple of numbers 7 and 17 within 5 multiples (1) 14, 42 (2) 14, 30 (3) 14, 28
The Correct Answer and Explanation is:
The common multiples of two numbers are those numbers that are divisible by both. To find the common multiples of 7 and 17, we first need to find their least common multiple (LCM).
Step 1: Find the LCM of 7 and 17
- Since 7 and 17 are both prime numbers, the LCM of 7 and 17 is simply their product:
LCM(7,17)=7×17=119\text{LCM}(7, 17) = 7 \times 17 = 119LCM(7,17)=7×17=119.
Step 2: Find the first few common multiples
The common multiples of 7 and 17 will be multiples of their LCM, so the multiples of 119 are:
- 119×1=119119 \times 1 = 119119×1=119
- 119×2=238119 \times 2 = 238119×2=238
- 119×3=357119 \times 3 = 357119×3=357
- 119×4=476119 \times 4 = 476119×4=476
- 119×5=595119 \times 5 = 595119×5=595
Step 3: Analyze the options provided
Now, let’s check the options:
- 14, 42: These are multiples of 7 (since 7×2=147 \times 2 = 147×2=14, and 7×6=427 \times 6 = 427×6=42), but they are not multiples of 17.
- 14, 30: These are also multiples of 7 (14 is 7×27 \times 27×2, and 30 is not a multiple of 7), but neither of these is a multiple of 17.
- 14, 28: These are multiples of 7, but they are not multiples of 17 either.
Step 4: Conclusion
None of the options provided (14, 42), (14, 30), or (14, 28) are common multiples of 7 and 17. The correct common multiples are based on multiples of the least common multiple of 7 and 17, which is 119. Therefore, there is no correct answer among the given options.
