Solutions: Sec 3.4 Playing with Digits

Page 60: Figure It Out

Digit Count Table:

• 1-digit: 9 (1 to 9)
• 2-digits: 90 (10 to 99)
• 3-digits: 900 (100 to 999)
• 4-digits: 9,000
• 5-digits: 90,000

Q1. Digit sum 14

• b. Smallest number: 59 ($5+9=14$).
• c. Largest 5-digit number: 95000.
• d. How big a number? You can make an infinitely large number by adding zeros (e.g., $5900000...$) or by adding 1s (e.g., $1111...$ fourteen times). The question asks “How big”, implying magnitude or length. You can always make a bigger number by adding more digits that sum to 0 (zeros) or distributing the sum into ones.

Q2. Digit sums from 40 to 70

• 40 (4), 41 (5), … 49 (13)
• 50 (5), …
• Observation: The digit sum increases by 1 until the number ends in 9, then drops. (e.g., $49 \to 13$, $50 \to 5$).

Q3. Consecutive digits (3-digit)

• $123 (6), 234 (9), 345 (12), 456 (15), 567 (18), 678 (21), 789 (24)$.
• Pattern: Multiples of 3.
• Reason: $(n) + (n+1) + (n+2) = 3n + 3 = 3(n+1)$. The sum is always divisible by 3.