Question

Show that every common multiple of two positive integers \(a\) and \(b\) is divisible by the least common multiple of \(a\) and \(b\).

Short answer

Question: Prove that every common multiple of two positive integers a and b is divisible by their least common multiple (LCM). Answer: In order to prove that every common multiple of two positive integers a and b is divisible by their LCM, we first defined the LCM as the smallest positive integer that is a multiple of both a and b. Then, we showed that the LCM is divisible by both a and b. We considered any common multiple of a and b, denoted by m, and used the divisibility property to prove that m is divisible by the LCM. Finally, we inferred that every common multiple of two positive integers a and b is indeed divisible by their LCM.

Step-by-step solution

Defining least common multiple (LCM)

In order to prove the given statement, we need to be clear on the definition of the LCM. The least common multiple (LCM) of two positive integers \(a\) and \(b\) is the smallest positive integer that is a multiple of both \(a\) and \(b\). Let's denote the LCM as \(l=\text{lcm}(a, b)\).

Show that LCM(a,b) is divisible by both a and b

Since \(l\) is a common multiple of \(a\) and \(b\), this means that \(l\) is divisible by \(a\) and \(b\). This can be written as: $$l=as=bt$$ where \(s\) and \(t\) are integers.

Consider any common multiple m

Now, let's consider any common multiple of \(a\) and \(b\), denoted by \(m\). Since \(m\) is a common multiple of \(a\) and \(b\), it can be written as follows: $$m=ax=by$$ where \(x\) and \(y\) are integers.

Using divisibility property to prove the statement

We want to prove that \(m\) is divisible by the LCM, \(l\). Using the definition of LCM, we know that \(l=as=bt\) (as mentioned in Step 2). We can rewrite this as: $$as=l=bt$$ Now, we can say that: $$a|l \text{ and } b|l$$ Using the divisibility property that if \(a|l\) and \(b|l\), then \(ab|l\), we can express this as: $$ab|lm$$ Since \(m\) is a common multiple, we can write: $$m=ax=by$$ Now, we can write the equation as follows: $$ab|lm=axm$$ From this, we can infer that: $$m(l-a)=0$$ Since \(m\) is a positive integer, this implies that \(a\) is divisible by \(l\), therefore \(m=k\cdot\text{lcm}(a,b)\) Thus, we have proved that every common multiple of two positive integers \(a\) and \(b\) is divisible by the least common multiple of \(a\) and \(b\).

Key concepts

Divisibility

Divisibility is an essential concept in mathematics that determines if one integer is a factor of another. In simple terms, when we say a number \( m \) is divisible by another number \( n \), it means \( m \) can be divided by \( n \) without leaving any remainder. For example, 10 is divisible by 2 because dividing 10 by 2 results in 5 with no remainder. To understand divisibility between numbers, it is important to grasp the division operation. When we write \( a|b \), this notation indicates that \( a \) divides \( b \) completely. This means there exists an integer \( k \) such that \( b = a \cdot k \). The idea of divisibility extends further when exploring factors and multiples of numbers, especially when dealing with least common multiples (LCM).A basic property of divisibility we often use is that if \( a|b \) and \( b|c \), then \( a|c \). This transitive property becomes useful in proving that common multiples (like LCM) are divisible by given numbers.

Common Multiples

Common multiples of two numbers are integers that are multiples of both numbers. For example, if we consider 3 and 4, their first few common multiples are 12, 24, and 36. Common multiples play a critical role in solving problems related to finding the least common multiple (LCM).To identify common multiples, let's denote two numbers as \( a \) and \( b \). A number \( m \) is a common multiple if it satisfies both \( a|m \) and \( b|m \). In other words, \( m \) should be divisible by both \( a \) and \( b \).Ultimately, among all common multiples, the least common multiple is the smallest positive integer that is a multiple of both \( a \) and \( b \). Hence, when considering any common multiple like \( m \), we can establish that \( m = k \cdot \text{lcm}(a, b) \) where \( k \) is an integer. This expression highlights that the LCM is a foundational building block from which other common multiples can be generated.

Integer Factorization

Integer factorization is the process of breaking down a number into its prime factors. Prime factors are the prime numbers that multiply together to give the original number. For instance, the integer factorization of 60 is \( 2^2 \times 3 \times 5 \).In the context of finding the least common multiple (LCM), integer factorization enables us to determine LCM efficiently by using the prime factorization of the involved numbers. The LCM can be found by taking the highest power of all prime factors appearing in the factorizations of the numbers.Consider two numbers \( a = 12 \) and \( b = 15 \). The prime factorization of 12 is \( 2^2 \times 3 \) and for 15 is \( 3 \times 5 \). The LCM of 12 and 15 would be \( 2^2 \times 3 \times 5 = 60 \), as we take the highest power of each prime that appears in their factorizations.This method ensures we're considering the smallest integer that is divisible by both \( a \) and \( b \), underscoring how integer factorization and LCM are intertwined in solving problems involving common multiples and divisibility.