Find q and r for the following pairs of positive integers a and b ,satisfying a= bq+ r. (1) a=13,b=3 (2) a=80,b=8 . From the above questions , what is the nature of q and r.?

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Solution

a = bq + r

(i) a= 13 and b = 3

13 = 3q + r Equation 1

(ii) a = 80 and b = 8

80 = 8q + r
Equation 2

Subtract equation 1 from 2

We will get

80-13 = 8q-3q

67 = 5q

q = 67/5 = 13.4

Put value of q in equation 1
We will get
13 = 3*13.4 + r

r = -40.2 +13 = -27.2

Answer : q= 13.4 and r = -27.2

q is positive and r is negative in nature

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