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Chapter 5: Problem 59

Find the least common multiple of the numbers. 16 and 42

Short Answer

Expert verified
The least common multiple (LCM) of 16 and 42 is 336.

Step by step solution

01

Prime Factorization of 16

Start by finding the prime factorization of 16. Prime number is a number that has only two positive divisors: 1 and the number itself. The number 16 can be divided by 2, leading to the result 8, which again can be divided by 2 to get 4, and finally by 2 again twice to get 1. So, the prime factorization of 16 is \(2^4\) or 2*2*2*2.
02

Prime Factorization of 42

Next, find the prime factorization of 42. The number 42 can be divided by 2, leading to the result 21. Then divide the resulting 21 by 3 to get 7, which is a prime number. So, the prime factorization of 42 is \(2*3*7\).
03

Find the LCM

The LCM is the product of the highest powers of all primes appearing in any of the factorizations. Here, from 16 we have the prime 2 raised to the power 4. From 42, we have the prime numbers 2, 3, and 7. The maximum power among the common primes is 2 (from 42), and the primes unique to 42 are 3 and 7. Multiplying all these together, we find \(2^4 * 3 * 7 = 336\) as the least common multiple of 16 and 42.

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