Exercise · ZW-ZERO-DIVISOR-008 · ★★☆ Intermediate
In \(\mathbb{Z}_{8}\), which pair with \(x\neq y\) satisfies \(4x=4y\)?
Choose one option.
Show answer
- A. \((x,y)=(1,3)\)
Explanation
With \(x=1\), \(y=3\): \(4\) and \(12\equiv4\). Cancellation fails because 4 is a zero divisor: \(4(x-y)=0\) with \(x-y\neq0\). In the other pairs the products differ.
Review the concept: Zero divisor
A zero divisor of a ring \(R\) is an element \(a\neq0\) for which there is another element \(b\neq0\) of \(R\) whose product with \(a\) is zero.
\[ab=0,\quad a\neq0,\ b\neq0\]
More practice problems: zero divisor
- ZW-ZERO-DIVISOR-001 In ℤ_12, which of these elements is a zero divisor?
- ZW-ZERO-DIVISOR-002 How many zero divisors does ℤ_10 have?
- ZW-ZERO-DIVISOR-003 Which of these rings has no zero divisors?
- ZW-ZERO-DIVISOR-004 In ℤ_15, which b≠0 satisfies 6b=0?
- ZW-ZERO-DIVISOR-005 In the ring of 2×2 real matrices, which of these matrices is a zero divisor?
- ZW-ZERO-DIVISOR-006 Which statement is true in a ring R with identity, R≠0?
- ZW-ZERO-DIVISOR-007 In ℤ×ℤ, which of these elements is a zero divisor?
- ZW-ZERO-DIVISOR-009 How many zero divisors does ℤ_21 have?