Chapter 1: Problem 4

Find the l.c.m. of the following: (a) 4,10 (b) \(16,30,40\) (c) \(15,16,25,32\)

Short Answer

Expert verified

a) (4, 10) b) (16, 30, 40) c) (15, 16, 25, 32) Answer: a) l.c.m.(4, 10) = 20 b) l.c.m.(16, 30, 40) = 240 c) l.c.m.(15, 16, 25, 32) = 2400

Step by step solution

(a) Find the prime factorization of each number

First, we need to find the prime factorization of each number in the set (4, 10). The prime factorization of 4 is \(2^2\) and the prime factorization of 10 is \(2^1 \times 5^1\).

(a) Determine the highest powers of each prime factor

Now, we need to find the highest power of each prime factor in the numbers. For this set of numbers, we have the prime factors 2 and 5. The highest power of 2 is \(2^2\) and the highest power of 5 is \(5^1\).

(a) Multiply the highest powers to find the l.c.m.

Finally, multiply the highest powers of each prime factor together to find the least common multiple. In this case, l.c.m.(4, 10) = \(2^2 \times 5^1 = 4 \times 5 = 20\).

(b) Find the prime factorization of each number

For the set (16, 30, 40), we need to find the prime factorization of each number: 16 = \(2^4\), 30 = \(2^1 \times 3^1 \times 5^1\), and 40 = \(2^3 \times 5^1\).

(b) Determine the highest powers of each prime factor

We need to find the highest power of each prime factor in the numbers: the highest power of 2 is \(2^4\), of 3 is \(3^1\), and of 5 is \(5^1\).

(b) Multiply the highest powers to find the l.c.m.

Now, we can multiply the highest powers of each prime factor to get the least common multiple: l.c.m.(16, 30, 40) = \(2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240\).

(c) Find the prime factorization of each number

For the set (15, 16, 25, 32), we need to find the prime factorization of each number: 15 = \(3^1 \times 5^1\), 16 = \(2^4\), 25 = \(5^2\), and 32 = \(2^5\).

(c) Determine the highest powers of each prime factor

We need to find the highest power of each prime factor in the numbers: the highest power of 2 is \(2^5\), of 3 is \(3^1\), and of 5 is \(5^2\).

(c) Multiply the highest powers to find the l.c.m.

Finally, multiply the highest powers of each prime factor to get the least common multiple: l.c.m.(15, 16, 25, 32) = \(2^5 \times 3^1 \times 5^2 = 32 \times 3 \times 25 = 2400\).

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Find the l.c.m. of the following: (a) 4,10 (b) \(16,30,40\) (c) \(15,16,25,32\)

Short Answer

Step by step solution

(a) Find the prime factorization of each number

(a) Determine the highest powers of each prime factor

(a) Multiply the highest powers to find the l.c.m.

(b) Find the prime factorization of each number

(b) Determine the highest powers of each prime factor

(b) Multiply the highest powers to find the l.c.m.

(c) Find the prime factorization of each number

(c) Determine the highest powers of each prime factor

(c) Multiply the highest powers to find the l.c.m.

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