After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4respectively. What will be the remainder if 84 divides the same number?a)80b)75c)41d)53Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Let consider 'a' as common multiple of the number
After the division of 'a' by 3 we get remainder 2
So, the number is 3a+2
but 'a' is also equal to 4a+1
So 3(4a+1) +2 = 12a + 5
and,
a is also equal to 7x+4
so 12(7a+4) +5
becomes 84a +48+5
i.e 84a + 53
hench when that Number is multiplied by '8a'remainder is 0 and +53
i.e 0+53 is 53

So, D is correct answer.

Thank you
Regards Pratik Wagh

Answer

Solution:

Let's assume the number to be x.

Given, x mod 3 = 2, x mod 4 = 1 and x mod 7 = 4.

We can solve this problem using the Chinese Remainder Theorem.

Step 1: Find the value of x mod 12.
- x mod 3 = 2 => x = 3k + 2, where k is an integer.
- x mod 4 = 1 => 3k + 2 mod 4 = 1 => k mod 4 = 3.
- k = 4n + 3, where n is an integer.
- Substituting the value of k in x = 3k + 2, we get x = 12n + 11.
- x mod 7 = 4 => 12n + 11 mod 7 = 4 => n mod 7 = 5.
- n = 7m + 5, where m is an integer.
- Substituting the value of n in x = 12n + 11, we get x = 84m + 83.
- Hence, x mod 12 = 11.

Step 2: Find the value of x mod 84.
- x mod 12 = 11 => x = 12k + 11, where k is an integer.
- Substituting the value of x in x mod 7 = 4, we get k mod 7 = 4.
- k = 7n + 4, where n is an integer.
- Substituting the value of k in x = 12k + 11, we get x = 84n + 59.
- Hence, the remainder when x is divided by 84 is 59.

Therefore, option (d) 53 is incorrect. The correct answer is (c) 41.

Answer

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

As the Number gives a remainder of 4 when it is divided by 7, then the number must be in form of (7x + 4)

The same gives remainder 1 when it is divided 4, so the number must be in the form of {4 × (7x + 4) + 1}

Also, the number when divided by 3 gives remainder 2, thus number must be in form of [3 × {4 × (7x + 4) + 1} + 2]

Now, On simplifying,
[3 × {4 × (7x + 4) + 1} + 2]
= 84x + 53
We get the final number 53 more than a multiple of 84 Hence, if the number is divided by 84,
The remainder will be 53

why is 7x+4 considered as the base?

Answer

Solution:

Let's assume the number as x.

When x is divided by 3, the remainder is 2.

So, x = 3a + 2, where a is some integer.

When x is divided by 4, the remainder is 1.

So, 3a + 2 = 4b + 1, where b is some integer.

Simplifying the above equation, we get:

a = (4b - 1)/3

The above equation tells us that (4b - 1) is divisible by 3.

When x is divided by 7, the remainder is 4.

So, 3a + 2 = 7c + 4, where c is some integer.

Simplifying the above equation, we get:

a = (7c + 2)/3

Substituting the value of a from equation (2) in equation (3), we get:

(4b - 1)/3 = (7c + 2)/3

4b - 1 = 7c + 2

4b = 7c + 3

The above equation tells us that (7c + 3) is divisible by 4.

Now, we need to find the remainder when x is divided by 84.

We know that 84 = 3 x 4 x 7.

So, we need to find the remainders when x is divided by 3, 4 and 7 separately, and then combine them to get the remainder when x is divided by 84.

Remainder when x is divided by 3 = 2

Remainder when x is divided by 4 = 1

Remainder when x is divided by 7 = 4

Let's find the remainder when x is divided by 84 using the Chinese Remainder Theorem.

We need to solve the following system of equations:

x ≡ 2 (mod 3)

x ≡ 1 (mod 4)

x ≡ 4 (mod 7)

We can use the following steps to solve this system of equations:

Step 1: Find M

M = 3 x 4 x 7 = 84

Step 2: Find M1, M2 and M3

M1 = 4 x 7 = 28

M2 = 3 x 7 = 21

M3 = 3 x 4 = 12

Step 3: Find the inverses of M1, M2 and M3

28^-1 ≡ 1 (mod 4)

21^-1 ≡ 3 (mod 7)

12^-1 ≡ 3 (mod 3)

Step 4: Find the values of x1, x2 and x3

x1 ≡ 2 (mod 3) => x1 = 2 + 3k1, where k1 is some integer

x1 ≡ 1 (mod 4) => 2 + 3k1 ≡ 1 (mod 4) => k1 ≡ 3 (mod 4) => k1 = 3 + 4k2, where k2 is some integer

x1 ≡ 4 (mod 7) => 2 + 3k1 = 4 + 28k3 =>

Answer

Answer

[{(7*1+4)*4+1}*3+2]=137
137/84=53(Reminder)

Answer

41

Answer

D

Answer

Since after division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively, the number is of form ((((4*4)+1)*3)+2)k = 53K.

Let k = 1; the number becomes 53

If it is divided by 84, the remainder is 53.

Therefore, the correct answer is Option D.

Answer

53

Answer

75

Answer

D is correct. obvious.

Answer

D) 53 is the option
(( 4*4)+1) *3) +1

Answer

D is correct

Similar UPSC Doubts

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2,1 and 4 respectively. What will be the remainder if 84 divides the same number?

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

After the division of a number successively by 3,4 and 7, the remainders obtained are 2,1 and 4 respectively. What will be the remainder if 84 divides the same number?

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