If a and b are positive integers, prove that;
ab = gcd(a,b)*lcm(a,b).
To prove the given statement:
ab = gcd(a,b) * lcm(a,b) where a and b are positive integers, we need to use the properties of greatest common divisor (gcd) and least common multiple (lcm).
First, let's define the greatest common divisor of a and b as gcd(a, b), and the least common multiple of a and b as lcm(a, b).
In order to prove the statement, we can make use of the fundamental property of the gcd and lcm:
Property 1: a * b = gcd(a, b) * lcm(a, b)
To prove this property, let's consider the prime factorization of both a and b.
Assume that the prime factorization of a is given by:
a = p₁^α₁ * p₂^α₂ * p₃^α₃ * ... * pₙ^αₙ
where p₁, p₂, p₃, ..., pₙ are prime factors of a, and α₁, α₂, α₃, ..., αₙ are their corresponding exponents.
Similarly, assume that the prime factorization of b is given by:
b = p₁^β₁ * p₂^β₂ * p₃^β₃ * ... * pₙ^βₙ
where p₁, p₂, p₃, ..., pₙ are prime factors of b, and β₁, β₂, β₃, ..., βₙ are their corresponding exponents.
Now, let's find the prime factorization of both the gcd(a,b) and lcm(a,b).
Since gcd(a,b) would only keep the common prime factors between a and b, the prime factorization of gcd(a,b) would be:
gcd(a, b) = p₁^min(α₁, β₁) * p₂^min(α₂, β₂) * p₃^min(α₃, β₃) * ... * pₙ^min(αₙ, βₙ)
Similarly, since lcm(a,b) would include all the prime factors of a and b, the prime factorization of lcm(a,b) would be:
lcm(a, b) = p₁^max(α₁, β₁) * p₂^max(α₂, β₂) * p₃^max(α₃, β₃) * ... * pₙ^max(αₙ, βₙ)
Now, we can calculate the product of gcd(a,b) and lcm(a,b).
gcd(a, b) * lcm(a, b) = (p₁^min(α₁, β₁) * p₂^min(α₂, β₂) * p₃^min(α₃, β₃) * ... * pₙ^min(αₙ, βₙ)) * (p₁^max(α₁, β₁) * p₂^max(α₂, β₂) * p₃^max(α₃, β₃) * ... * pₙ^max(αₙ, βₙ))
By using the properties of exponents, we can simplify this expression as:
gcd(a, b) * lcm(a, b) = p₁^(min(α₁, β₁) + max(α₁, β₁)) * p₂^(min(α₂, β₂) + max(α₂, β₂)) * p₃^(min(α₃, β₃) + max(α₃, β₃)) * ... * pₙ^(min(αₙ, βₙ) + max(αₙ, βₙ))
Now, let's analyze the exponents. For each prime factor pᵢ, we have two cases:
Case 1: αᵢ < βᵢ
In this case, min(αᵢ, βᵢ) = αᵢ and max(αᵢ, βᵢ) = βᵢ
Case 2: αᵢ ≥ βᵢ
In this case, min(αᵢ, βᵢ) = βᵢ and max(αᵢ, βᵢ) = αᵢ
In both cases, min(αᵢ, βᵢ) + max(αᵢ, βᵢ) = αᵢ + βᵢ
Substituting this back into the equation, we get:
gcd(a, b) * lcm(a, b) = p₁^(α₁ + β₁) * p₂^(α₂ + β₂) * p₃^(α₃ + β₃) * ... * pₙ^(αₙ + βₙ)
Now, notice that α₁ + β₁ is the exponent of p₁ in the prime factorization of a * b. The same holds for all other primes pᵢ.
Therefore, we can conclude that:
gcd(a,b) * lcm(a,b) = a * b
Hence, we have proven that
ab = gcd(a,b) * lcm(a,b) for positive integers a and b.