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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?
- a)80
- b)75
- c)41
- d)53
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
After the division of a number successively by 3, 4 and 7, the remaind...
In the successive division, the quotient of first division becomes the dividend of the second division and so on.
Let the last quotient be p, so the last dividend will be 7p + 4 which is the quotient of the second division.
So, the second dividend is (7p + 4) � 4 + 1.
Applying the same logic, the number = 3 {4(7p + 4) + 1} + 2 = 84p + 53
Hence, if the number is divided by 84, the remainder is 53.
Let the last quotient be p, so the last dividend will be 7p + 4 which is the quotient of the second division.
So, the second dividend is (7p + 4) � 4 + 1.
Applying the same logic, the number = 3 {4(7p + 4) + 1} + 2 = 84p + 53
Hence, if the number is divided by 84, the remainder is 53.
Hence Option D is correct
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Most Upvoted Answer
After the division of a number successively by 3, 4 and 7, the remaind...
Given:
- Remainders when the number is divided by 3, 4, and 7 successively are 2, 1, and 4 respectively.
To find:
- The remainder when the number is divided by 84.
Solution:
- Let's assume the number to be x.
- From the given information, we can write three equations:
- x ≡ 2 (mod 3)
- x ≡ 1 (mod 4)
- x ≡ 4 (mod 7)
- To solve these equations, we can use the Chinese Remainder Theorem (CRT):
- Let M = 3 x 4 x 7 = 84 (product of moduli)
- Let M1 = 4 x 7 = 28, M2 = 3 x 7 = 21, and M3 = 3 x 4 = 12 (factors of M)
- Let y1, y2, and y3 be the inverses of M1, M2, and M3 modulo 3, 4, and 7 respectively.
- Then the solution of the system of equations is given by:
- x ≡ (2 x 28 x y1) + (1 x 21 x y2) + (4 x 12 x y3) (mod M)
- To find the inverses, we can use the Extended Euclidean Algorithm:
- y1 ≡ 1 (mod 3), y1 ≡ 0 (mod 4), y1 ≡ 0 (mod 7) => y1 = 1
- y2 ≡ 0 (mod 3), y2 ≡ 1 (mod 4), y2 ≡ 0 (mod 7) => y2 = 5
- y3 ≡ 2 (mod 3), y3 ≡ 1 (mod 4), y3 ≡ 1 (mod 7) => y3 = 23
- Substituting the values of y1, y2, and y3 in the formula, we get:
- x ≡ (2 x 28 x 1) + (1 x 21 x 5) + (4 x 12 x 23) (mod 84)
- x ≡ 56 + 105 + 1104 (mod 84)
- x ≡ 81 (mod 84)
- Therefore, the remainder when the number is divided by 84 is 81, which is option (d).
Answer: d) 53
- Remainders when the number is divided by 3, 4, and 7 successively are 2, 1, and 4 respectively.
To find:
- The remainder when the number is divided by 84.
Solution:
- Let's assume the number to be x.
- From the given information, we can write three equations:
- x ≡ 2 (mod 3)
- x ≡ 1 (mod 4)
- x ≡ 4 (mod 7)
- To solve these equations, we can use the Chinese Remainder Theorem (CRT):
- Let M = 3 x 4 x 7 = 84 (product of moduli)
- Let M1 = 4 x 7 = 28, M2 = 3 x 7 = 21, and M3 = 3 x 4 = 12 (factors of M)
- Let y1, y2, and y3 be the inverses of M1, M2, and M3 modulo 3, 4, and 7 respectively.
- Then the solution of the system of equations is given by:
- x ≡ (2 x 28 x y1) + (1 x 21 x y2) + (4 x 12 x y3) (mod M)
- To find the inverses, we can use the Extended Euclidean Algorithm:
- y1 ≡ 1 (mod 3), y1 ≡ 0 (mod 4), y1 ≡ 0 (mod 7) => y1 = 1
- y2 ≡ 0 (mod 3), y2 ≡ 1 (mod 4), y2 ≡ 0 (mod 7) => y2 = 5
- y3 ≡ 2 (mod 3), y3 ≡ 1 (mod 4), y3 ≡ 1 (mod 7) => y3 = 23
- Substituting the values of y1, y2, and y3 in the formula, we get:
- x ≡ (2 x 28 x 1) + (1 x 21 x 5) + (4 x 12 x 23) (mod 84)
- x ≡ 56 + 105 + 1104 (mod 84)
- x ≡ 81 (mod 84)
- Therefore, the remainder when the number is divided by 84 is 81, which is option (d).
Answer: d) 53
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Community Answer
After the division of a number successively by 3, 4 and 7, the remaind...
In the successive division, the quotient of first division becomes the dividend of the second division and so on.
Let the last quotient be p, so the last dividend will be 7p + 4 which is the quotient of the second division.
So, the second dividend is (7p + 4) × 4 + 1.
Applying the same logic, the number = 3 {4(7p + 4) + 1} + 2 = 84p + 53
Hence, if the number is divided by 84, the remainder is 53.
Let the last quotient be p, so the last dividend will be 7p + 4 which is the quotient of the second division.
So, the second dividend is (7p + 4) × 4 + 1.
Applying the same logic, the number = 3 {4(7p + 4) + 1} + 2 = 84p + 53
Hence, if the number is divided by 84, the remainder is 53.
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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?a)80b)75c)41d)53Correct answer is option 'D'. Can you explain this answer?
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