Question 9 of 25intermediate🔧 ApplyNumerical3 marks

Find the smallest multiple of 9 with no odd digits.

Correct Answer

Given: Find the smallest multiple of 9 with no odd digits. Rule: A multiple of 9 must have digits summing to a multiple of 9. Available even digits: 0, 2, 4, 6, 8

Solution: Single-digit: 9 has odd digit. Two-digit: Need sum = 9 or 18 Sum = 9: Not possible (9 is odd) Sum = 18: Smallest is 288 (2+8+8=18) Checking: 18, 36, 54, 72, 90 all have odd digits. Three-digit: 108, 126, 144, etc. all have odd digits.

Verification: 288 ÷ 9 = 32 All digits are even.

Answer: 288

Exercise: Figure it Out (Section 2) | Q: 2 | (Page Page 15)
For More Understanding

Explanation

The chapter explains divisibility by 9. A multiple of 9 must have digits summing to a multiple of 9. For no odd digits, we can only use 0, 2, 4, 6, 8. Single-digit multiples of 9: 9 (has odd digit). Two-digit multiples with even digits: need sum = 9 or 18. For sum = 9 with even digits: not possible (9 is odd). For sum = 18: smallest is 288 (2+8+8=18). Checking smaller numbers: 18, 36, 54, 72, 90 all have odd digits. Three-digit candidates: 108, 126, 144, etc. all have odd digits. The smallest is 288.

Solution Steps

  1. Step 1: A multiple of 9 must have digit sum divisible by 9

  2. Step 2: Even digits available: 0, 2, 4, 6, 8

  3. Step 3: Single-digit: 9 has odd digit

  4. Step 4: Two-digit: smallest sum = 9 (odd, impossible), next sum = 18

  5. Step 5: For sum 18 with even digits: 2+8+8=18 gives 288

  6. Step 6: Verify: 288 ÷ 9 = 32, all digits even, smallest such number

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