1.

find q and r for the following pairs of positive integers a and b satisfying` a = bq+r``(1)a =13 , b=13 ``(2)a=8 , b=20``(3)a=125 , b=5``(4)a=132 ,b=11`

Answer» given that, ` a= b*q +r `
in option 1 , given that `a= 13, b=13 `
`13= 13*q + r`
`13*(1-q) = r`
`if q= 0, r= -13`
`if q=1, r= 0`
`if q= 2 , r=-13` not possible
`:.` 2 solutions can be possible.
for option 2. given that `a= 8 , b=20`
`8= 20*q+ r`
`4(2-5*q)= r`
`if q= 0 . r=8`
`if q=1, r=-12 ` not possible
so. 1 solution is possible.
`a= 125 & b= 5`
`125 = 5*q + r`
`5*(25-q) = r`
`if q=0, r=125`
`if q=1, r=24*5`
`if q=2, r= 23*5`
`..`
`..`
`..`
`..`
`if q=25, r= 0`
`if q= 26 , r=-ve`
so, 25 solutions are possible.`a= 132, b=11`
`132= 11*q + r`
`132 - 11*q = r`
`11*(12- q) = r`
`q= 0 , r= 12*11`
`q=1 , r= 11*11`
`q=2 , r = 10*11`
`..`
`..`
`q=12, r=0`
`q= 13 , r= -ve`
so, 12 solutions are possible.


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