Question 36 of 58beginner💡 UnderstandShort Answer2 marks

What are the possible remainders at each step when dividing by 7, and why can the remainder not be 0?

Correct Answer

When dividing by 7, the possible remainders at each step are 1, 2, 3, 4, 5, or 6. The remainder cannot be 0 because if a remainder of 0 is reached, the division process terminates and the decimal expansion stops. However, for fractions like 1/7 that result in a repeating decimal, the division never reaches a remainder of 0. Because there is a limited number of possible remainders (only 1 through 6), a remainder must eventually show up a second time, causing the entire division process to loop infinitely.

Exercise: 3.6.1 Rational Decimals: Terminating and Repeating | Q: 1 | (Page 18)
For More Understanding

Explanation

The textbook explains that when performing long division by 7, the only possible remainders are 1 through 6. It prompts the student to question why 0 is excluded. The context clarifies that a remainder of 0 means the decimal terminates (stops). Since the discussion centers on rational numbers with repeating decimal representations (like 1/7), the remainder cannot be 0; otherwise, the decimal would not repeat. The limited number of non-zero remainders guarantees that a remainder will eventually repeat, creating the looping cycle of a repeating decimal.

Solution Steps

  1. Step 1: Identify the possible remainders when dividing by 7, which are 1, 2, 3, 4, 5, and 6.

  2. Step 2: Explain why 0 is not possible in this context: a remainder of 0 means the division terminates and the decimal stops.

  3. Step 3: Connect the non-zero remainders to the concept of repeating decimals: since remainders are limited to 1-6, one must eventually repeat, causing the division process to loop.

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