Solution to Exercise 1.20:

`;;; Normal Order`

```
```; remainder count: 0

(gcd 206 40)

; rc: 0

(if (= 40 0)

206

(gcd 40 (remainder 206 40)))

; rc: 0

(gcd 40 (remainder 206 40))

; rc: 0

(if (= (remainder 206 40) 0)

40

(gcd (remainder 206 40)

(remainder 40 (remainder 206 40))))

; rc: 1

(if (= 6 0)

40

(gcd (remainder 206 40)

(remainder 40 (remainder 206 40))))

; rc: 1

(gcd (remainder 206 40)

(remainder 40 (remainder 206 40)))

; rc: 1

(if (= (remainder 40 (remainder 206 40)) 0)

(remainder 206 40)

(gcd (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))

; rc: 2

(if (= (remainder 40 6) 0)

(remainder 206 40)

(gcd (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))

; rc: 3

(if (= 4 0)

(remainder 206 40)

(gcd (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))

; rc: 3

(gcd (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))

; rc: 3

(if (= (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

0)

(remainder 40 (remainder 206 40))

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))))

; rc: 4

(if (= (remainder 6

(remainder 40

(remainder 206 40)))

0)

(remainder 40 (remainder 206 40))

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))))

; rc: 5

(if (= (remainder 6

(remainder 40

6))

0)

(remainder 40 (remainder 206 40))

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))))

; rc: 6

(if (= (remainder 6 4) 0)

(remainder 40 (remainder 206 40))

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))))

; rc: 7

(if (= 2 0)

(remainder 40 (remainder 206 40))

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40 (remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))))

; rc: 7

(gcd (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))

; rc: 7

(if (= (remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))

0)

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(gcd (remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))

(remainder (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))))

; rc: 14 (there were 7 calls to remainder in the = form above)

(if (= 0 0)

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(gcd (remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40))))

(remainder (remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

(remainder (remainder 40

(remainder 206 40))

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))))))

; rc: 14

(remainder (remainder 206 40)

(remainder 40

(remainder 206 40)))

; rc: 15

(remainder 6

(remainder 40

(remainder 206 40)))

; rc: 16

(remainder 6 (remainder 40 6))

; rc: 17

(remainder 6 4)

`; rc: 18`

2

For normal order, remainder is performed 18 times.

`;;; Applicative Order`

```
```; rc: 0

(gcd 206 40)

; rc: 0

(if (= 40 0)

206

(gcd 40 (remainder 206 40)))

; rc: 0

(gcd 40 (remainder 206 40))

; rc: 1

(gcd 40 6)

; rc: 1

(if (= 6 0)

40

(gcd 6 (remainder 40 6)))

; rc: 1

(gcd 6 (remainder 40 6))

; rc: 2

(gcd 6 4)

; rc: 2

(if (= 4 0)

6

(gcd 4 (remainder 6 4)))

; rc: 2

(gcd 4 (remainder 6 4))

; rc: 3

(gcd 4 2)

; rc: 3

(if (= 2 0)

4

(gcd 2 (remainder 4 2)))

; rc: 3

(gcd 2 (remainder 4 2))

; rc: 4

(gcd 2 0)

; rc: 4

(if (= 0 0)

2

(gcd 0 (remainder 2 0)))

`; rc: 4`

2

For applicative order, remainder is only performed four times.