If a divides c and b divides c , then a b divides c .

The statement "If \( a \) divides \( c \) and \( b \) divides \( c \), then \( ab \) divides \( c \)" is not always true. It depends on the relationship between \( a \) and \( b \).

Here’s why:

### Basic Definitions:
- \( a \) divides \( c \) (written as \( a | c \)) means that \( c = a \cdot k \) for some integer \( k \).
- Similarly, \( b | c \) means \( c = b \cdot m \) for some integer \( m \).

However, \( ab \) divides \( c \) does **not** always follow directly from this. The divisibility of \( ab \) by \( c \) depends on the greatest common divisor (gcd) of \( a \) and \( b \).

### Counterexample:
Take \( a = 2 \), \( b = 4 \), and \( c = 8 \).
- \( 2 \) divides \( 8 \) (since \( 8 = 2 \times 4 \)),
- \( 4 \) divides \( 8 \) (since \( 8 = 4 \times 2 \)),
- But \( 2 \times 4 = 8 \) does not divide \( 8 \times 2 = 16 \), meaning \( ab \) does **not** always hold.

### True Statement:
If \( a \) and \( b \) divide \( c \), then \( \text{lcm}(a, b) | c \), where lcm stands for least \( a \cdirected accurately "debugg

If a divides c and b divides c, then ab divides c.

Explore the concept of divisibility in number theory with a detailed explanation of why 'if a divides c and b divides c, then ab divides c' is not always true. Learn about the relationship between greatest common divisor (gcd) and least common multiple (lcm) with practical examples. Suitable for students in Grades 9-12.

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