QUESTION 20 How many 3-digit numbers can be formed from the digits 0, 3, 4, 5, 7, 8 and 9,...

A number is divisible by 4 if its last two digits form a number divisible by 4. We need to find all valid 2-digit endings (using distinct digits from our set) that are divisible by 4, then count how many valid first digits (non-zero, distinct) can be placed before each ending. The total count across all valid endings is 28.

Step-by-step Derivation:
Step 1: Identify the divisibility rule for 4.
A number is divisible by 4 if its last two digits form a number divisible by 4.

Step 2: Find all 2-digit endings from {0, 3, 4, 5, 7, 8, 9} that are divisible by 4.
We need XY where XY ≡ 0 (mod 4) and X ≠ Y.

Checking systematically:

• 0_: 00 (invalid, repeated), 03, 04 (✓), 05, 07, 08 (✓), 09
• 3_: 30, 32, 33 (invalid, no 2), 34, 35, 36 (no 6), 37, 38, 39
• 4_: 40 (✓), 43, 44 (invalid, repeated), 45, 47, 48 (✓), 49
• 5_: 50, 52, 53, 54, 55, 56, 57, 58, 59
• 7_: 70, 71, 72, 73, 74, 75, 76, 77, 78, 79
• 8_: 80 (✓), 83, 84 (✓), 85, 87, 88 (invalid, repeated), 89
• 9_: 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Valid 2-digit endings divisible by 4 from our digits (no repeats):
04, 08, 40, 48, 80, 84, 20 (no 2), 32 (no 2), 36 (no 6), 44 (repeated), 52 (no 2), 56 (no 6), 60 (no 6), 64 (no 6), 68 (no 6), 72 (no 2), 76 (no 6), 92 (no 2), 96 (no 6)

Actual valid endings from {0, 3, 4, 5, 7, 8, 9}:

• 04 ✓, 08 ✓, 40 ✓, 48 ✓, 80 ✓, 84 ✓
• 03÷4=0.75 ✗, 05÷4=1.25 ✗, 07÷4=1.75 ✗, 09÷4=2.25 ✗
• 30÷4=7.5 ✗, 34÷4=8.5 ✗, 35÷4=8.75 ✗, 37÷4=9.25 ✗, 38÷4=9.5 ✗, 39÷4=9.75 ✗
• 43÷4=10.75 ✗, 45÷4=11.25 ✗, 47÷4=11.75 ✗, 49÷4=12.25 ✗
• 50÷4=12.5 ✗, 53÷4=13.25 ✗, 54÷4=13.5 ✗, 57÷4=14.25 ✗, 58÷4=14.5 ✗, 59÷4=14.75 ✗
• 70÷4=17.5 ✗, 73÷4=18.25 ✗, 74÷4=18.5 ✗, 75÷4=18.75 ✗, 77÷4=19.25 ✗, 78÷4=19.5 ✗, 79÷4=19.75 ✗
• 83÷4=20.75 ✗, 85÷4=21.25 ✗, 87÷4=21.75 ✗, 89÷4=22.25 ✗
• 90÷4=22.5 ✗, 93÷4=23.25 ✗, 94÷4=23.5 ✗, 95÷4=23.75 ✗, 97÷4=24.25 ✗, 98÷4=24.5 ✗, 99÷4=24.75 ✗

Valid 2-digit endings: {04, 08, 40, 48, 80, 84}

Step 3: For each ending, count valid first digits (non-zero, distinct from the two used).
Available digits: {0, 3, 4, 5, 7, 8, 9} (7 total)

• Ending 04: First digit ∈ {3, 5, 7, 8, 9} (exclude 0 as first digit, 0 and 4 used) → 5 numbers
• Ending 08: First digit ∈ {3, 4, 5, 7, 9} (exclude 0 as first digit, 0 and 8 used) → 5 numbers
• Ending 40: First digit ∈ {3, 5, 7, 8, 9} (exclude 0 as first digit, 4 and 0 used) → 5 numbers
• Ending 48: First digit ∈ {3, 5, 7, 9, 0} but 0 can't be first → {3, 5, 7, 9} → 4 numbers
• Ending 80: First digit ∈ {3, 4, 5, 7, 9} (exclude 0 as first digit, 8 and 0 used) → 5 numbers
• Ending 84: First digit ∈ {3, 5, 7, 9, 0} but 0 can't be first → {3, 5, 7, 9} → 4 numbers

Total: 5 + 5 + 5 + 4 + 5 + 4 = 28