Final answer:
The least common denominator for the fractions a/2, b/3, c/9, and d/15 is obtained by finding the least common multiple of the denominators, which are 2, 3, 9, and 15. The LCM is b. 90, which makes 90 the least common denominator.
Step-by-step explanation:
The question asks us to find the least common denominator (LCD) for the given fractions a/2, b/3, c/9, and d/15. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder. To find the LCD, we look for the least common multiple (LCM) of the denominators, which are 2, 3, 9, and 15.
Firstly, we list the prime factors of each denominator:
2 is prime.
3 is prime.
9 = 3 × 3.
15 = 3 × 5.
To find the LCM, we take the highest powers of all prime factors that appear in any of the numbers:
The highest power of 2 is 2¹, as it appears in 2.
The highest power of 3 is 3², as it appears in 9.
The highest power of 5 is 5¹, as it appears in 15.
By multiplying these together, we get the LCM: 2¹ × 3² × 5¹ = 2 × 9 × 5 = 18 × 5 = 90. Therefore, the LCD is 90.