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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)41
- b)80
- c)75
- d)53
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
After the division of a number successively by 3, 4 and 7, the remain...
To find the remainder when 84 divides the same number, we need to analyze the given information.
Given:
- The number gives a remainder of 2 when divided by 3.
- The number gives a remainder of 1 when divided by 4.
- The number gives a remainder of 4 when divided by 7.
To find the number, we can set up a system of congruences using the Chinese Remainder Theorem.
Step 1: Set up the congruences
Let the number be represented by x. We can set up the following congruences:
x ≡ 2 (mod 3)
x ≡ 1 (mod 4)
x ≡ 4 (mod 7)
Step 2: Solve the congruences
To solve these congruences, we can use the method of successive substitution.
From the first congruence, we can write x = 3a + 2, where a is an integer.
Substituting this into the second congruence, we have:
3a + 2 ≡ 1 (mod 4)
3a ≡ -1 ≡ 3 (mod 4)
a ≡ 1 (mod 4)
So, a can be written as a = 4b + 1, where b is an integer.
Substituting this into the first congruence, we have:
x = 3(4b + 1) + 2
x = 12b + 5
Substituting this into the third congruence, we have:
12b + 5 ≡ 4 (mod 7)
12b ≡ -1 ≡ 6 (mod 7)
b ≡ 6 (mod 7)
So, b can be written as b = 7c + 6, where c is an integer.
Substituting this into the previous equation, we have:
x = 12(7c + 6) + 5
x = 84c + 77
Step 3: Find the remainder when 84 divides the number
To find the remainder when 84 divides the number, we can substitute x = 84c + 77 into the congruences:
x ≡ 77 (mod 3)
x ≡ 77 (mod 4)
x ≡ 77 (mod 7)
Since 77 is congruent to 5 (mod 3), 1 (mod 4), and 0 (mod 7), the remainder when 84 divides the number will also be 5.
Therefore, the correct answer is option D) 53.
Given:
- The number gives a remainder of 2 when divided by 3.
- The number gives a remainder of 1 when divided by 4.
- The number gives a remainder of 4 when divided by 7.
To find the number, we can set up a system of congruences using the Chinese Remainder Theorem.
Step 1: Set up the congruences
Let the number be represented by x. We can set up the following congruences:
x ≡ 2 (mod 3)
x ≡ 1 (mod 4)
x ≡ 4 (mod 7)
Step 2: Solve the congruences
To solve these congruences, we can use the method of successive substitution.
From the first congruence, we can write x = 3a + 2, where a is an integer.
Substituting this into the second congruence, we have:
3a + 2 ≡ 1 (mod 4)
3a ≡ -1 ≡ 3 (mod 4)
a ≡ 1 (mod 4)
So, a can be written as a = 4b + 1, where b is an integer.
Substituting this into the first congruence, we have:
x = 3(4b + 1) + 2
x = 12b + 5
Substituting this into the third congruence, we have:
12b + 5 ≡ 4 (mod 7)
12b ≡ -1 ≡ 6 (mod 7)
b ≡ 6 (mod 7)
So, b can be written as b = 7c + 6, where c is an integer.
Substituting this into the previous equation, we have:
x = 12(7c + 6) + 5
x = 84c + 77
Step 3: Find the remainder when 84 divides the number
To find the remainder when 84 divides the number, we can substitute x = 84c + 77 into the congruences:
x ≡ 77 (mod 3)
x ≡ 77 (mod 4)
x ≡ 77 (mod 7)
Since 77 is congruent to 5 (mod 3), 1 (mod 4), and 0 (mod 7), the remainder when 84 divides the number will also be 5.
Therefore, the correct answer is option D) 53.
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Community Answer
After the division of a number successively by 3, 4 and 7, the remain...
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.
Option d) is the correct answer.
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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)41b)80c)75d)53Correct answer is option 'D'. Can you explain this answer?
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