The lowest common multiple of 4 and 14 is 28, the smallest positive number that can be divided evenly by both 4 and 14. Because of that, in other words, 28 is divisible by 4 with a remainder of 0, and it is also divisible by 14 with a remainder of 0. This result can be confirmed using lists of multiples, prime factorization, or the relationship between the greatest common divisor and the least common multiple. Understanding how to find it is useful because the least common multiple, often shortened to LCM, appears in fractions, scheduling, grouping, repeating patterns, and many other mathematical tasks That's the part that actually makes a difference..
What Is a Least Common Multiple?
A multiple of a number is any number that can be produced by multiplying that number by a whole number. The multiples of 4 include:
- 4
- 8
- 12
- 16
- 20
- 24
- 28
- 32
- 36
- 40
A common multiple of 4 and 14 is a number that appears in the multiple lists of both numbers. The least common multiple is the smallest positive number that is a multiple of both values Easy to understand, harder to ignore..
Don't overlook the word least. Even so, it carries more weight than people think. Although 56 is also a common multiple of 4 and 14, it is not the lowest common multiple because 28 comes first Simple as that..
Method 1: Compare Lists of Multiples
The most direct way to find the lowest common multiple of 4 and 14 is to write out the multiples of each number until a shared value appears.
Multiples of 4
[ 4, 8, 12, 16, 20, 24, 28, 32, \ldots ]
Multiples of 14
[ 14, 28, 42, 56, 70, \ldots ]
The first number that appears in both lists is 28. Therefore:
[ \boxed{\operatorname{LCM}(4,14)=28} ]
This method is easy to understand and works especially well for small numbers. For larger numbers, however, writing long lists may be inefficient. Prime factorization is usually faster.
Method 2: Use Prime Factorization
Prime factorization breaks a number down into its basic prime factors. The least common multiple is then found by selecting the highest power of every prime that appears in either factorization.
Factor 4:
[ 4=2\times2=2^2 ]
Factor 14:
[ 14=2\times7 ]
Now list the prime factors:
| Number | Prime Factorization |
|---|---|
| 4 | (2^2) |
| 14 | (2^1\times7^1) |
For each prime, select the greatest exponent:
- For 2, the greatest exponent is 2, giving (2^2).
- For 7, the only exponent is 1, giving (7^1).
Multiply the selected prime powers:
[ \operatorname{LCM}(4,14)=2^2\times7=4\times7=28 ]
The same answer is obtained. This is a reliable method because it ensures that the result is divisible by every prime factor required by either number.
Method 3: Use the GCD-LCM Relationship
Two numbers have a useful connection involving their greatest common divisor, or GCD, and their least common multiple:
[ a\times b=\gcd(a,b)\times\operatorname{LCM}(a,b) ]
The greatest common divisor of 4 and 14 is 2 because 2 is the largest number that divides both without a remainder Worth knowing..
Rearranging the formula gives:
[ \operatorname{LCM}(a,b)=\frac{a\times b}{\gcd(a
