Remainders: Part-2
Type: Difference between divisor and remainder is same
On dividing by p, q and r the number gives different remainders a, b and c where
p-a=q-b=r-c
Let’s take an example to understand this

Example
What is the minimum two-digit number which when divided by 3, 4, 5 leaves 1, 2, 3 as the remainder respectively?
Solution:
Let P be the required two digit number .As given in the question
P = 3 d + 1 =3 (d + 1) -2
P = 4 e + 2= 4( e + 1) -2
P = 5 f + 3= 5 (f+1) – 2

In each of above cases we observe difference between divisor and remainder is same.
Hence we have same remainder -2 when divided by 3,4,5
Thus number N must be in the form L.C.M. (3,4,5) k – 2 =60 k – 2
Since we are looking for two digit number putting k =1 we get 58 as minimum two-digit number which when divided by 3, 4, 5 leaves 1, 2, 3 as the remainder respectively.

Example
What is the minimum three-digit number which when divided by 2, 5, 6   leaves 1, 4, 5 as the remainder respectively?
Solution:
Let P be the required three digit number .As given in the question
P = 2 d + 1 =2 (d + 1) -1
P = 5 e + 4 = 5( e + 1) -1
P = 6 f + 5 = 6 (f+1) – 1
In each of above cases we observe difference between divisor and remainder is same.
Hence we have same remainder -1 when divided by 2,5,6
Thus number N must be in the form L.C.M. (2,5,6) k – 1 =30 k – 1
Since we are looking for two digit number putting k =34 we get 1019 as minimum three-digit number which when divided by 2, 5, 6 leaves 1, 4, 5 as the remainder respectively.

Assignment:
QUESTIONS:
1. What is the remainder when 72 + 51 is divided by 8 ?
a)1
b)2
c)3
d)5
Solution : Since 72 is multiple of 8 hence it leaves remainder zero and 51 leaves remainder 3 when divided by 8.Hence 72 + 51 leaves remainder 3 when divided by 8.

2. What is the remainder when 111 + 68 + 48 is divided by 11 ?
a)5
b)7
c)9
d)None
Solution:
111 leaves remainder 1 when divided by 11
68 leaves remainder 2 when divided by 11
48 leaves remainder 4 when divided by 11
Thus 111 + 68 + 48 leaves remainder 1+2+4 =7 when divided by 11
3. What is the remainder when 42×56 is divided by 8 ?
a)0
b)1
c)2
d)3
Solution :
56 is a multiple of 8 hence remainder is zero.

4.What is the remainder when 48×65 is divided by 7 ?
a)1
b)2
c)3
d)5
48 leaves remainder 6 and 65 leaves remainder 2 so 48 65 divided by 7 gives 6×2=12 but 12 is greater than 7 hence we further divide by 7 and we obtain 5 as remainder.

5. What is the remainder of 84x71x41 by 9?
a)2
b)3
c)6
d)8
84 leaves remainder 3, 71 leaves remainder 8 and 41 leaves remainder 5 when divided by 9. So 84x71x42 divided by 9 gives 3 8 =120 but 120 is greater than 9 hence we further divide by 9 and we obtain 3 as remainder.

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