Question 8 of 19intermediate🔧 ApplyNumerical3 marks

Find the smallest multiple of 9 with no odd digits.

Correct Answer

Step 1: A number with no odd digits can only have digits 0, 2, 4, 6, 8.

Step 2: For divisibility by 9, sum of digits must be divisible by 9.

Step 3: Check multiples of 9 in order: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, ...

Step 4: All these have at least one odd digit until we reach 288.

Step 5: 288 has digits 2, 8, 8 (all even) and sum = 18 (divisible by 9).

Step 6: 288 ÷ 9 = 32, confirming it is a multiple of 9.

Answer: The smallest multiple of 9 with no odd digits is 288.

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

Explanation

A number with no odd digits can only contain the digits 0, 2, 4, 6, 8. For a number to be divisible by 9, the sum of its digits must be divisible by 9. Checking multiples of 9: 9 (contains 9), 18 (contains 1), 27 (contains 7), 36 (contains 3), 45 (contains 5), 54 (contains 5), 63 (contains 3), 72 (contains 7), 81 (contains 1), 90 (contains 9), 99 (contains 9), 108 (contains 1), and so on. The first multiple of 9 with all even digits is 288 (digits: 2, 8, 8; sum = 18, divisible by 9).

Solution Steps

  1. Step 1: A number with no odd digits can only have digits 0, 2, 4, 6, 8

  2. Step 2: For divisibility by 9, sum of digits must be divisible by 9

  3. Step 3: Check multiples of 9 in order: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, ...

  4. Step 4: All these have at least one odd digit until we reach 288

  5. Step 5: 288 has digits 2, 8, 8 (all even) and sum = 18 (divisible by 9)

  6. Step 6: 288 ÷ 9 = 32, confirming it is a multiple of 9

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