Question

Q (25): The remainder on dividing given integers a and b by 7 are respectively 5 and
4. Then, the remainder when ab is divided by 7 is
(a) 5
(b) 4
(c)0
(d) 6​

Answer 1

Answer:

Again using the basic identity of a division, this means that when (A - B) is divided by 7, the quotient is (p - q - 1) and the remainder is 5. As seen above, if we divide A-B by 7, the remainder will be 5. a-b = -2 hence its reminder we Take the smallest nonnegative number congruent to -2 modulo 7 i.e. 5.

Explaination:

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Answer 2

The remainder when ab is divided by 7 is 6 [ The correct option (d) 6 ]

Given :

The remainder on dividing given integers a and b by 7 are respectively 5 and 4

To find :

The remainder when ab is divided by 7 is

(a) 5

(b) 4

(c) 0

(d) 6

Concept :

Euclid's Division Lemma :

For given two positive integers a and b, there exist unique integers q and r such that

a = bq + r where 0 ≤ r < b

Solution :

Step 1 of 2 :

Find the expression for a and b

Here it is given that the remainder on dividing given integers a and b by 7 are respectively 5 and 4

Using Euclid's Division Lemma we get

a = 7p + 5 where p is an integer

b = 7q + 4 where p is an integer

Step 2 of 2 :

Find the remainder when ab is divided by 7

$\sf ab$

$\displaystyle \sf = (7p + 5)(7q + 4)$

$\displaystyle \sf = 7p(7q + 4) + 5(7q + 4)$

$\displaystyle \sf = 49pq + 28p + 35q + 20$

$\displaystyle \sf = 7(7pq + 4p + 5q + 2) + 6$

Now , 7(7pq + 4p + 5q + 2) is completely divisible by 7

∴ The remainder when ab is divided by 7 is 6

Hence the correct option (d) 6

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