**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

Answer – c

**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

answer – b

**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

answer – a

**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

Answer – 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

answer-b

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.

1019 as minimum three-digit ?

k =34?