The smallest 8-digit number with exactly five different digits is 10,000,234. The smallest 8-digit number is 10,000,000. It already uses two different digits: 1 and 0. To keep the number as small as possible while adding exactly three more new digits, we introduce the smallest available digits — 2, 3, and 4 — and place them at the rightmost positions. This gives 10,000,234, which uses digits {0, 1, 2, 3, 4} — five distinct digits.
Smallest 8-digit number: 10,000,000.
Smallest 8-digit number with exactly 5 different digits: 10,000,234.
Digits used: {0, 1, 2, 3, 4} — five distinct digits.
Strategy: keep the leftmost digits unchanged (1,0,0,0,0) and add smallest new digits (2,3,4) at the rightmost positions.
Placing larger digits rightward minimises the number.
10,000,234 in words: one crore, two hundred and thirty-four.
Goal: Find the smallest 8-digit number using exactly 5 different digit values.
Step 1: What is the smallest 8-digit number? → 10,000,000 (the first 8-digit number — starts with 1, rest are zeros)
Step 2: How many different digits does 10,000,000 use? → Digits: 1, 0, 0, 0, 0, 0, 0, 0 → Different digits: {0, 1} = only 2 different digits → We need 5 different digits, so we need 3 more.
Step 3: What are the smallest digits we can add? → We already have 0 and 1. → Next smallest available: 2, 3, 4
Step 4: Where should we place them to keep the number smallest? → To minimise the number, place new (larger) digits as far to the RIGHT as possible. → The leftmost positions should remain 1, 0, 0, 0, 0 (unchanged). → Place 2, 3, 4 in the last three positions (positions 6, 7, 8 from left).
Step 5: Build the number: → Positions: 1 _ _ _ _ _ _ _ → Fill with zeros: 1, 0, 0, 0, 0, _, _, _ → Add 2, 3, 4 at end (ascending order to minimise): 1, 0, 0, 0, 0, 2, 3, 4 → Number: 10,000,234
Step 6: Verify: → Is it 8 digits? 10,000,234 → Yes ✓ → Different digits: 1, 0, 0, 0, 0, 2, 3, 4 → unique digits = {0, 1, 2, 3, 4} = 5 different digits ✓ → Is it the smallest possible? Yes — placing 2, 3, 4 earlier would make the number larger. ✓
Answer: 10,000,234
Why not place the new digits earlier in the number?
Comparison: • 10,000,234 — new digits at positions 6, 7, 8 (rightmost) • 10,002,034 — 2 moved earlier → larger number • 10,023,400 — digits moved even earlier → larger number • 10,234,000 — digits at positions 3–5 → even larger
Rule: To minimise a number, keep larger digits as far right as possible.
Why not use different new digits like 2, 3, 5? • {0, 1, 2, 3, 4} gives smaller digits than {0, 1, 2, 3, 5} or {0, 1, 2, 4, 5} • Using 2, 3, 4 at the end gives: 10,000,234 • Using 2, 3, 5 at the end gives: 10,000,235 → larger • Using 2, 4, 5 at the end gives: 10,000,245 → larger
Therefore, {0, 1, 2, 3, 4} placed as 10,000,234 is the smallest 8-digit number with exactly five different digits.
10,000,234 in place value form:
Position | Value | Digit Crore (10,000,000) | 1 × 1,00,00,000 | 1 10 Lakh (1,000,000) | 0 × 10,00,000 | 0 1 Lakh (100,000) | 0 × 1,00,000 | 0 10 Thousand (10,000) | 0 × 10,000 | 0 1 Thousand (1,000) | 0 × 1,000 | 0 Hundreds (100) | 2 × 100 | 2 Tens (10) | 3 × 10 | 3 Ones (1) | 4 × 1 | 4
10,000,234 = One crore and two hundred thirty-four.
Different digits used: 0, 1, 2, 3, 4 → 5 different digits ✓
In Indian number system: 1,00,00,234 (one crore two hundred thirty-four)
The smallest 8-digit number with five different digits is 10,000,234. Explanation: The smallest 8-digit number is 10,000,000 (uses digits {0,1} — only 2 different). We need 3 more different digits; the smallest available are 2, 3, 4. To keep the number small, place them at the rightmost positions: 10,000,234. Verification: digits are 1,0,0,0,0,2,3,4 → distinct digits: {0,1,2,3,4} = 5 ✓
Strategy: (1) Start with the smallest number in that digit range (e.g., 10,000,000 for 8-digit). (2) Check how many different digits it already uses. (3) Identify the smallest missing digits needed to reach the required count. (4) Place those new digits at the rightmost positions to keep the number as small as possible. (5) Arrange the new digits in ascending order at those positions.
The smallest 8-digit number is 10,000,000 (one crore). It uses only 2 different digits: 1 and 0.
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