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In dividing a number by 1001, a student employed the method of short division. He divided the number successively by 7, 11 and 13 and got the remainders 4, 6, 6 respectively. What would have been the remainder if he had divided the number by 1001?
- a)16
- b)508
- c)144
- d)576
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In dividing a number by 1001, a student employed the method of short d...
Let “p” be the number,
Suppose,
p ÷ 7 = q, remainder 4
q ÷ 11 = r, remainder 6
r ÷ 13 = s, remainder 6
Hence,
r = 13s +6
q = 11r + 6
⇒ 11(13s+6)+6 = 143s+72
p = 7q + 4
⇒ 7 (143s + 72) + 4
⇒ 1001s + 504 + 4
⇒ 1001s + 508
Therefore, if P is divided by 1001, we will get 508 as remainder.
Suppose,
p ÷ 7 = q, remainder 4
q ÷ 11 = r, remainder 6
r ÷ 13 = s, remainder 6
Hence,
r = 13s +6
q = 11r + 6
⇒ 11(13s+6)+6 = 143s+72
p = 7q + 4
⇒ 7 (143s + 72) + 4
⇒ 1001s + 504 + 4
⇒ 1001s + 508
Therefore, if P is divided by 1001, we will get 508 as remainder.
Most Upvoted Answer
In dividing a number by 1001, a student employed the method of short d...
Given information:
- The student divided the number successively by 7, 11, and 13 and got the remainders 4, 6, and 6 respectively.
- The final step is dividing the number by 1001.
To find the remainder when dividing by 1001, we can use the Chinese Remainder Theorem. This theorem states that if we have a system of congruences in the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
...
x ≡ an (mod mn)
where m1, m2, ..., mn are pairwise coprime (i.e., they have no common factors), then there exists a unique solution for x modulo (m1 * m2 * ... * mn).
In this case, we can set up the following system of congruences:
x ≡ 4 (mod 7)
x ≡ 6 (mod 11)
x ≡ 6 (mod 13)
Solving this system of congruences will give us the remainder when dividing the number by 1001.
Solving the System of Congruences:
We can solve the system of congruences using the Chinese Remainder Theorem.
1. Find the product of all moduli: M = 7 * 11 * 13 = 1001.
2. Find the individual moduli: M1 = 1001/7 = 143, M2 = 1001/11 = 91, M3 = 1001/13 = 77.
3. Find the inverses of the individual moduli with respect to their respective moduli:
- For 143, the inverse is 5 because 143 * 5 ≡ 1 (mod 7) (i.e., 143 * 5 divided by 7 leaves a remainder of 1).
- For 91, the inverse is 10 because 91 * 10 ≡ 1 (mod 11).
- For 77, the inverse is 12 because 77 * 12 ≡ 1 (mod 13).
4. Multiply the remainders by the respective moduli and their inverses:
- For 4 (mod 7), the term is 4 * 143 * 5 = 2860.
- For 6 (mod 11), the term is 6 * 91 * 10 = 5460.
- For 6 (mod 13), the term is 6 * 77 * 12 = 5544.
5. Add up the terms obtained in step 4: 2860 + 5460 + 5544 = 13864.
6. Take the result from step 5 modulo the product of all moduli: 13864 ≡ 508 (mod 1001).
Therefore, the remainder when dividing the number by 1001 is 508. Hence, option B is the correct answer.
- The student divided the number successively by 7, 11, and 13 and got the remainders 4, 6, and 6 respectively.
- The final step is dividing the number by 1001.
To find the remainder when dividing by 1001, we can use the Chinese Remainder Theorem. This theorem states that if we have a system of congruences in the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
...
x ≡ an (mod mn)
where m1, m2, ..., mn are pairwise coprime (i.e., they have no common factors), then there exists a unique solution for x modulo (m1 * m2 * ... * mn).
In this case, we can set up the following system of congruences:
x ≡ 4 (mod 7)
x ≡ 6 (mod 11)
x ≡ 6 (mod 13)
Solving this system of congruences will give us the remainder when dividing the number by 1001.
Solving the System of Congruences:
We can solve the system of congruences using the Chinese Remainder Theorem.
1. Find the product of all moduli: M = 7 * 11 * 13 = 1001.
2. Find the individual moduli: M1 = 1001/7 = 143, M2 = 1001/11 = 91, M3 = 1001/13 = 77.
3. Find the inverses of the individual moduli with respect to their respective moduli:
- For 143, the inverse is 5 because 143 * 5 ≡ 1 (mod 7) (i.e., 143 * 5 divided by 7 leaves a remainder of 1).
- For 91, the inverse is 10 because 91 * 10 ≡ 1 (mod 11).
- For 77, the inverse is 12 because 77 * 12 ≡ 1 (mod 13).
4. Multiply the remainders by the respective moduli and their inverses:
- For 4 (mod 7), the term is 4 * 143 * 5 = 2860.
- For 6 (mod 11), the term is 6 * 91 * 10 = 5460.
- For 6 (mod 13), the term is 6 * 77 * 12 = 5544.
5. Add up the terms obtained in step 4: 2860 + 5460 + 5544 = 13864.
6. Take the result from step 5 modulo the product of all moduli: 13864 ≡ 508 (mod 1001).
Therefore, the remainder when dividing the number by 1001 is 508. Hence, option B is the correct answer.
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In dividing a number by 1001, a student employed the method of short division. He divided the number successively by 7, 11 and 13 and got the remainders 4, 6, 6 respectively. What would have been the remainder if he had divided the number by 1001?a)16b)508c)144d)576Correct answer is option 'B'. Can you explain this answer?
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