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Chapter 5: Problem 58

Find the least common multiple of the numbers. 25 and 70

Short Answer

Expert verified
The least common multiple of 25 and 70 is 350.

Step by step solution

01

Prime Factorization of 25

We have to break 25 into its prime factors. This is done by dividing 25 by the smallest prime number (other than 1), which is 2. Since 25 is not divisible by 2, we move to the next smallest prime number, which is 3, and so on, until we find a number that can divide 25. This is 5. Therefore the prime factorisation of 25 is \(5 \times 5\) or \(5^2\).
02

Prime Factorization of 70

Now we do the same for 70. Starting with 2, we see that 70 can be divided by 2 to get 35. Next, 35 can be divided by 5 to get 7. At this point, we have broken down 70 to all its prime factors, which are \(2, 5, 7\). Therefore the prime factorisation of 70 can be expressed as \(2 \times 5 \times 7\).
03

Find the Least Common Multiple (LCM)

To find the LCM, we need to multiply together the highest powers of all the prime factors that appear in either 25 or 70. So, the LCM is \(2^1 \times 5^2 \times 7^1 = 350\).

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,-6,6,-6, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(5,15,45,135, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{7}\), when \(a_{1}=4, r=2\)

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,10,16,22, \ldots\)

What is the common difference in an arithmetic sequence?
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