Divisibility And Remainder

To find the least common multiple (LCM) of 9, 12, and 15, we first find the prime factorizations of the numbers:
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Step 1: LCM Calculation
The LCM is obtained by taking the highest power of each prime that appears in the factorizations:
- For , the highest power is (from 12)
- For , the highest power is (from 9)
- For , the highest power is (from 15)
Thus, the LCM is:
Thus, the LCM of 9, 12, and 15 is .

We are given two expressions: Step 1: The highest common factor (HCF) is found by taking the lowest power of each common factor in both expressions. Since are distinct prime numbers, we examine the powers of each factor in both expressions. - For , the powers are and , so the lowest power is . - For , the powers are and , so the lowest power is . - For , the powers are and , so the lowest power is . Step 2: Therefore, the HCF is: Thus, the HCF of and is .