Question

A number when divided by 7 leaves a
remainder 3 and the resulting
quotient, when divided by 11 leaves a
remainder 6. If the same number when
divided by 11 leaves a remainder m
and the resulting quotient when
divided by 7 leaves a remainder n.
What are the values of m and n,
respectively?
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Answer 1

A number (x) when divided by 7 leaves a remainder 3

According to remainder theorem, x = 7q +3……………. (1)

the resulting quotient when divided by 11 leaves a remainder 6

So, q = 11b + 6………………... (2)

From (1) and (2)

x = 7(11b +6) + 3=77b+45

when x divided by 11 leaves a reminder m

⇒ 7b + 4 + (1/11) ⇒ Remainder = +1

Clearly, we can see that when x is divided by 11 remainder is 1.

So, m = 1

Quotient = 7b + 4

resulting quotient when divided by 7 leaves a remainder n

⇒ b + (4/7) ⇒ Remainder = +4

So, n = 4

Answer 2

A number (x) when divided by 7 leaves a remainder 3

According to remainder theorem, x = 7q +3……………. (1)

the resulting quotient when divided by 11 leaves a remainder 6

So, q = 11b + 6………………... (2)

From (1) and (2)

x = 7(11b +6) + 3=77b+45

when x divided by 11 leaves a reminder m

x/11 = 77b+ 45/ 11 = 77b+ 44+1/ 11

⇒ 7b + 4 + (1/11) ⇒ Remainder = +1

Clearly, we can see that when x is divided by 11 remainder is 1.

So, m = 1

Quotient = 7b + 4

resulting quotient when divided by 7 leaves a remainder n

7b+ 4/ 7

⇒ b + (4/7) ⇒ Remainder = +4

So, n = 4