Euler's Totient Function φ is multiplicative for coprime integers. Which statement is true?

Euler's Totient Function φ is multiplicative for coprime integers. Which statement is true?

• A. φ(ab) = φ(a) φ(b) if gcd(a,b) = 1
• B. φ(ab) = φ(a) + φ(b) always
• C. φ(ab) = φ(a) φ(b) for all a,b
• D. φ(n) is always even

Answer: (A)

The key idea is how Euler’s totient counts numbers that are coprime to a given modulus and how those counts behave when you combine two moduli that share no common factors. If a and b are coprime, every number less than ab that is coprime to ab corresponds uniquely to a pair: a residue that is coprime to a and a residue that is coprime to b. Because these choices are independent, you multiply the counts, giving φ(ab) = φ(a) φ(b) when gcd(a, b) = 1. That makes the statement true: the totient function is multiplicative over coprime integers, so φ(ab) equals φ(a) times φ(b) under the coprime condition. Why the other ideas don’t hold generally: φ(ab) = φ(a) + φ(b) would mix counts in a way that isn’t correct for coprime moduli. And φ(ab) = φ(a) φ(b) for all a and b fails when a and b share factors; for example, if a = b = 2, φ(4) = 2 but φ(2) φ(2) = 1, which shows the product rule requires gcd(a, b) = 1. Finally, φ(n) being always even isn’t true: φ(2) = 1, and φ(1) = 1, so there are odd totients.

The key idea is how Euler’s totient counts numbers that are coprime to a given modulus and how those counts behave when you combine two moduli that share no common factors. If a and b are coprime, every number less than ab that is coprime to ab corresponds uniquely to a pair: a residue that is coprime to a and a residue that is coprime to b. Because these choices are independent, you multiply the counts, giving φ(ab) = φ(a) φ(b) when gcd(a, b) = 1.

That makes the statement true: the totient function is multiplicative over coprime integers, so φ(ab) equals φ(a) times φ(b) under the coprime condition.

Why the other ideas don’t hold generally: φ(ab) = φ(a) + φ(b) would mix counts in a way that isn’t correct for coprime moduli. And φ(ab) = φ(a) φ(b) for all a and b fails when a and b share factors; for example, if a = b = 2, φ(4) = 2 but φ(2) φ(2) = 1, which shows the product rule requires gcd(a, b) = 1. Finally, φ(n) being always even isn’t true: φ(2) = 1, and φ(1) = 1, so there are odd totients.