Question

List five integers that are congruent to 4 modulo 12.

Short answer

The five integers that are congruent to 4 modulo 12 are 4, 16, 28, 40, and 52.

Step-by-step solution

Concept of Division Algorithm 

• \(a\)divides \(b\)if there exists an integer\(c\) such that \(b = ac\)Notation:\(a\mid b\)
• Let \(a\) be an integer and \(d\) be a positive integer.
• Then there are unique Integers \(q\) and \(T\)with \(0 \le \;r < d\)such that \(a = dq + r\).
• \(q\)is called the quotient and \(T\)is called the remainder.
• \(\begin{array}{*{20}{l}}{q = a{\rm{ div }}d}\\{\;r = a{\rm{ mod }}d}\end{array}\)
• \(a\) is congruent to \(b\) modulo \(m\) if \(m\) divides \(a - b\) Notation : \(a \equiv b\;(\,\bmod \,\;m)\)

List the five integers that are congruent to 4 modulo 12

We need to find five integers that are congruent to 4 modulo 12

By the definition of congruent modulo 12, \(x\) is congruent to 4 modulo 12, when 12

divides \(x - 4\)

Since 12 divides 0, 12, 24, 36, 48, …… possible values for \(x - 4\) are 0, 12, 28, 40, 52, ….

\(x - 4 = 0 \Rightarrow x = 4\)

\(x - 4 = 12 \Rightarrow x = 16\)

\(x - 4 = 24 \Rightarrow x = 28\)

\(x - 4 = 36 \Rightarrow x = 40\)

\(x - 4 = 48 \Rightarrow x = 52\)

Hence the five integers that are congruent to 4 modulo 12 are 4, 16, 28, 40, and 52.