In Exercise, find the decimal expansion of the given rational number. All these expansions...

In Exercise, find the decimal expansion of the given rational number. All these expansions are too long to fit in a calculator but can be readily found by using the hint.

283/47 [Hint: The first part of dividing 1 by 17 involves working this division problem: 1,000,000 ÷ 17. The method of Exercise 1 shows that the quotient is 58,823 and the remainder is 9. Thus the decimal expansion of 1/ 17 begins .058823, and the next block of digits in the expansion will be the quotient in the problem 9,000,000 ÷ 17. The remainder when 9,000,000 is divided by 17 is 13, so the next block of digits in the expansion of 1/ 17 is the quotient in the problem 13,000,000 ÷ 17. Continue in this way until the decimal expansion repeats.]

Exercise 1

Finding remainders with a calculator

If you use long division to divide 369 by 7, you obtain:

If you use a calculator to find 369 ÷ 7, the answer is displayed as 52.71428571. Observe that the integer part of this calculator answer, 52, is the quotient when you do the problem by long division. The usual “checking procedure” for long division shows that

 7 · 52 + 5 = 369 or, equivalently 369 − 7 · 52 = 5.

Thus, the remainder is

Use this method to find the quotient and remainder in these problems:

(a) 5683 ÷ 9

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(b) 1,000,000 ÷ 19

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(c) 53,000,000 ÷ 37