Question 17 of 31beginner🔧 ApplyShort Answer2 marks

If 31z5 is a multiple of 9, where z is a digit, what is the value of z? Explain why there are two answers to this problem.

Correct Answer

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.

Given: Number = 31z5, where z is a digit (0-9)

Step 1: Sum of digits = 3 + 1 + z + 5 = 9 + z

Step 2: For divisibility by 9: 9 + z must be divisible by 9

Step 3: Possible values:

  • 9 + z = 9 → z = 0
  • 9 + z = 18 → z = 9

Answer: z = 0 or z = 9

Why there are two answers: Since z is a single digit (0-9), the sum 9 + z can only equal 9 or 18 to be divisible by 9. This gives two valid values for z.

Exercise: Figure it Out (Section 4) | Q: 1 | (Page Page 21)
For More Understanding

Explanation

The chapter explains the divisibility rule for 9: a number is divisible by 9 if and only if the sum of its digits is divisible by 9. For 31z5, the sum is 3 + 1 + z + 5 = 9 + z. For this to be divisible by 9, we need 9 + z = 9 or 9 + z = 18 (since z is a single digit from 0 to 9). This gives z = 0 or z = 9.

Solution Steps

  1. Sum of digits = 3 + 1 + z + 5 = 9 + z

  2. For divisibility by 9: 9 + z must be divisible by 9

  3. Possible values: 9 + z = 9 (z = 0) or 9 + z = 18 (z = 9)

  4. Both 3105 and 3195 are divisible by 9

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