Answer
See explanation
Work Step by Step
Below is a direct derivation from the definition of McCarthy’s 91 function:
\[
M(n) \;=\;
\begin{cases}
n-10, & \text{if } n>100,\\[6pt]
M\bigl(M(n+11)\bigr), & \text{if } n\le 100.
\end{cases}
\]
We want to show:
1. \(M(86) = M(91)\).
2. \(M(91) = 91\).
---
## 1. Showing \(M(86) = M(91)\)
Since \(86 \le 100\), the definition says
\[
M(86) \;=\; M\bigl(M(86+11)\bigr) \;=\; M\bigl(M(97)\bigr).
\]
So we must evaluate \(M(97)\). Again \(97 \le 100\), so
\[
M(97) \;=\; M\bigl(M(97+11)\bigr) \;=\; M\bigl(M(108)\bigr).
\]
But \(108 > 100\), so
\[
M(108) \;=\; 108 - 10 \;=\; 98.
\]
Hence
\[
M(97) \;=\; M(98).
\]
Similarly, \(98 \le 100\), so
\[
M(98) \;=\; M\bigl(M(98+11)\bigr) \;=\; M\bigl(M(109)\bigr),
\]
and \(109 > 100 \implies M(109) = 99\). So
\[
M(98) = M(99).
\]
Continuing in the same fashion:
- \(99 \le 100 \implies M(99) = M(M(110))\),
- \(110 > 100 \implies M(110) = 100\),
- so \(M(99) = M(100)\).
Then:
- \(100 \le 100 \implies M(100) = M(M(111))\),
- \(111 > 100 \implies M(111) = 101\),
- so \(M(100) = M(101)\).
Finally:
- \(101 > 100 \implies M(101) = 91\).
Hence working backward:
\[
M(100) = 91,\quad M(99) = 91,\quad M(98) = 91,\quad M(97) = 91.
\]
Therefore
\[
M(86) \;=\; M\bigl(M(97)\bigr) \;=\; M(91).
\]
This completes the proof of part (a).
---
## 2. Showing \(M(91) = 91\)
We do a similar chain. Since \(91 \le 100\):
\[
M(91) \;=\; M\bigl(M(91+11)\bigr) \;=\; M\bigl(M(102)\bigr).
\]
But \(102 > 100\), so \(M(102) = 102 - 10 = 92\). Hence
\[
M(91) = M(92).
\]
Next \(92 \le 100 \implies M(92) = M(M(103))\), and \(103 > 100 \implies M(103)=93\). So
\[
M(92) = M(93).
\]
Repeating this pattern:
1. \(93 \le 100\)
\(\;M(93) = M(M(104))\),
\(104 > 100 \implies M(104)=94.\)
So \(M(93)=M(94).\)
2. \(94 \le 100\)
\(\;M(94)=M(M(105))\),
\(105>100 \implies M(105)=95.\)
So \(M(94)=M(95).\)
3. \(95 \le 100\)
\(\;M(95)=M(M(106))\),
\(106>100 \implies M(106)=96.\)
So \(M(95)=M(96).\)
4. \(96 \le 100\)
\(\;M(96)=M(M(107))\),
\(107>100 \implies M(107)=97.\)
So \(M(96)=M(97).\)
5. \(97 \le 100\)
\(\;M(97)=M(M(108))\),
\(108>100 \implies M(108)=98.\)
So \(M(97)=M(98).\)
6. \(98 \le 100\)
\(\;M(98)=M(M(109))\),
\(109>100 \implies M(109)=99.\)
So \(M(98)=M(99).\)
7. \(99 \le 100\)
\(\;M(99)=M(M(110))\),
\(110>100 \implies M(110)=100.\)
So \(M(99)=M(100).\)
8. \(100 \le 100\)
\(\;M(100)=M(M(111))\),
\(111>100 \implies M(111)=101.\)
So \(M(100)=M(101).\)
9. \(101 > 100\)
\(\;M(101)=91.\)
Tracing back up shows
\[
M(100)=91,\quad M(99)=91,\quad M(98)=91,\quad \dots,\quad M(92)=91,\quad M(91)=91.
\]
Thus \(M(91)=91\), completing part (b).
---
## Final Answers
1. **\(M(86) = M(91)\)**.
2. **\(M(91) = 91\).**
Consequently, it also follows that \(M(86)=91\).
