Lecture 05 — Numbers, Remainders, and Cycles

Learning objectives

• Use integer division and remainder to model cycles
• Reason about divisibility and digit sums
• Recognise when modulo simplifies a problem

Case-based lecture: you attempt, we discuss, then we reveal.

The plan

• Four cases: cs-017, cs-018, cs-019, cs-020
• Each: 5-minute attempt → discussion → reveal → prompt
• Worked examples in between — ask anything
• Break mid-lecture · exit question at the end

Key definitions

• Integer division `//` — division keeping only the whole part: `17 // 5` is 3.
• Remainder (modulo) `%` — what is left after whole divisions: `17 % 5` is 2.
• Cycle — a pattern repeating every k steps; `n % k` names the position inside the cycle.
• Digit extraction — `n % 10` is the last digit of n; `n // 10` removes it.
• Divisor — d divides n exactly when `n % d == 0`.
• Simulation — stepping a process rule by rule, as it would happen, instead of computing a formula.

The pager schedule as a cycle

Expected: 2 then 5

Cost: O(1) for both variants - arithmetic beats simulating day by day.

Digits, sums, and the divisible-by-3 rule

n = 4723
total = 0
while n > 0:
    total += n % 10
    n //= 10
print(total)

Expected: 16 then False

Cost: O(d) where d is the digit count (about log10 of n) - logarithmic in the value, which surprises students.

• **Input:** 5 people, rotation starts with person 1 on day 1; day 47
• **Output:** day 47 -> ((47-1) mod 5)+1 = person 2; with weekend skip: count only weekdays
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-017 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Propose one extra rule that would change the verdict — is it an improvement?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-017 — not embedded in this deck.

• **Input:** 4723
• **Output:** digit sum 16 -> not divisible by 3; peel: 4723 % 10 = 3, then 472, ...
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-018 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which assumption would you attack first if you had to argue the other side?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-018 — not embedded in this deck.

• **Input:** dead digit 4, N = 30
• **Output:** 1,2,3,5,6,7,8,9,10,11,12,13,15,... (23 numbers below 30 avoid the digit 4)
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-019 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Propose one extra rule that would change the verdict — is it an improvement?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-019 — not embedded in this deck.

• **Input:** n = 7 people, k = 3
• **Output:** survivor position 4 (0-indexed 3)
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-020 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which assumption would you attack first if you had to argue the other side?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-020 — not embedded in this deck.

Common misconceptions

• Treating % as trivia to memorise instead of a cycle-position tool.
• Starting cycles at 1 vs 0 - state the convention before computing.
• Extracting digits with strings when the arithmetic is the point of the exercise.
• Simulating the Josephus ring for large n instead of tabulating small cases and finding the pattern.

Summary & exit question

• Named technique today: Numbers, Remainders, and Cycles
• Four cases attempted — methods before answers

Exit question (choose one, 60 seconds)

1. Compute 17 mod 5 and 17 // 5, and say what each means for a 5-day cycle.
2. Why is remainder better than an if-chain for schedules?
3. What replaced a loop in the Josephus analysis?