Exercise · ZW-DIVISIBILITY-010 · ★★★ Advanced
What is the smallest positive integer \(n\) such that \(12\mid n^2\)?
Choose one option.
Show answer
- C. 6
Explanation
\(12=2^2\cdot3\) requires \(n^2\) to contain a 3, so \(3\mid n\), and \(2^2\), so \(2\mid n\). Hence \(6\mid n\) and \(6^2=36=12\cdot3\). 12 also works but is not the smallest.
Review the concept: Divisibility
An integer \(a\neq0\) divides an integer \(b\), written \(a\mid b\), when \(b\) is an integer multiple of \(a\).
\[a\mid b\iff b=ak\ \text{for some } k\in\mathbb{Z}\]
More practice problems: divisibility
- ZW-DIVISIBILITY-001 Which of the following statements is true?
- ZW-DIVISIBILITY-002 How many positive multiples of 6 lie between 1 and 100?
- ZW-DIVISIBILITY-003 If a| b and b| c, what always follows?
- ZW-DIVISIBILITY-004 Which of these numbers is divisible by 12?
- ZW-DIVISIBILITY-005 How many positive divisors does 36 have?
- ZW-DIVISIBILITY-006 How many positive integers n satisfy n| n+12?
- ZW-DIVISIBILITY-007 What is the largest positive integer a that divides both 24 and 36?
- ZW-DIVISIBILITY-008 Which of the following statements is false?