Question

List all integers between -100 and 100 that are congruent to -1 modulo 25 .

Short answer

\(a = \left\{ { - 1, - 26, - 51, - 76,{\rm{ }}24,{\rm{ }}49,{\rm{ }}74,{\rm{ }}99} \right\}\)

Step-by-step solution

Concept of Division Algorithm 

• 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.
• \(\begin{array}{*{20}{l}}{q = a{\rm{ div }}d}\\{\;r = a{\rm{ mod }}d}\end{array}\)

Listing all integers between -100 and 100 that are congruent to -1 modulo 25

\(a \equiv - 1\,\;\bmod \,\;25\)

We can find all values of \(a\) such that \( - 100 \le a \le 100\) by consecutively

Subtracting/adding 25 from -1 until we obtain all values between -100 and 100

We first start by consecutively subtracting:

\(\begin{array}{c}a \equiv - 1\,\;\bmod \,\;25\\ \equiv - 1 - 25\,\;\bmod \,\;25\\ \equiv - 26\;\,\bmod \,\;25\\ \equiv - 26 - 25\;\,\bmod \,\;25\end{array}\)

\(\begin{array}{c} \equiv - 51\,\;\bmod \;\,25\\ \equiv - 51 - 25\;\,\bmod \;\,25\\ \equiv - 76\;\,\bmod \,\;25\\ \equiv - 76 - 25\,\;\bmod \,\;25\\ \equiv - 101\,\;\bmod \,\;25\end{array}\)

Since -101 is the first value not in\( - 100 \le a \le 100\), \(a\)can then take on all previous

Values in the above derivation:

\(a = \{ - 76, - 51, - 26, - 1\} \)

Note: this set is not complete yet, we still need to determine all integers above -1.

Determining all integers above -1

We next start by consecutively adding:

\(\begin{array}{c}a \equiv - 1\;\,\bmod \;\,25\\ \equiv - 1 + 25\,\;\bmod \,\;25\\ \equiv 24\,\;\bmod \,\;25\\ \equiv 24 + 25\;\,\bmod \,\;25\\ \equiv 49\,\;\bmod \;\,25\end{array}\)

\(\begin{array}{c} \equiv 49 + 25\;\,\bmod \,\;25\\ \equiv 74\,\;\bmod \;\,25\\ \equiv 74 + 25\;\,\bmod \,\;25\\ \equiv 99\;\,\bmod \;\,25\\ \equiv 99 + 25\,\;\bmod \;\,25\\ \equiv 124\;\,\bmod \,\;25\end{array}\)

Since 124 is the first value not in\( - 100 \le a \le 100\), \(a\)can then take on all previous

Values in the above derivation:

\(a = \{ - 1,24,49,74,99\} \)

Combining this set with the set obtained using consecutive subtractions:

\(a = \left\{ { - 1, - 26, - 51, - 76,{\rm{ }}24,{\rm{ }}49,{\rm{ }}74,{\rm{ }}99} \right\}\)

Hence all integers between -100 and 100 are\(a = \left\{ { - 1, - 26, - 51, - 76,{\rm{ }}24,{\rm{ }}49,{\rm{ }}74,{\rm{ }}99} \right\}\)