The Greatest Number Which Divides 281 And 1249, Leaving The Reminder, Five And Seven Respectively Is Option A 23 Option B2 76 Option C1 38 Option D 69

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Introduction

In mathematics, the greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. In this article, we will discuss how to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively.

Understanding the Problem

To solve this problem, we need to find the greatest common divisor (GCD) of 281 and 1249. The GCD is the largest number that divides both numbers without leaving a remainder. However, in this case, we are given that the remainder is five and seven respectively. This means that we need to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively.

Finding the Greatest Common Divisor (GCD)

To find the GCD of 281 and 1249, we can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly applying the division algorithm.

Step 1: Divide 1249 by 281

To start the Euclidean algorithm, we divide 1249 by 281. This gives us a quotient of 4 and a remainder of 125.

Step 2: Divide 281 by 125

Next, we divide 281 by 125. This gives us a quotient of 2 and a remainder of 31.

Step 3: Divide 125 by 31

We continue the Euclidean algorithm by dividing 125 by 31. This gives us a quotient of 4 and a remainder of 1.

Step 4: Divide 31 by 1

Finally, we divide 31 by 1. This gives us a quotient of 31 and a remainder of 0.

Finding the Greatest Number

Since the remainder is 0, we know that 1 is the greatest common divisor (GCD) of 281 and 1249. However, we are given that the remainder is five and seven respectively. This means that we need to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively.

Using the Chinese Remainder Theorem

To solve this problem, we can use the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if we have a system of congruences:

x ≡ a1 (mod n1) x ≡ a2 (mod n2)

where n1 and n2 are coprime, then there is a unique solution modulo n1n2.

Step 1: Define the Congruences

We define the congruences:

x ≡ 5 (mod 281) x ≡ 7 (mod 1249)

Step 2: Find the Product of the Moduli

We find the product of the moduli:

n1n2 = 281 * 1249

Step 3: Find the Modular Multiplicative Inverse

We find the modular multiplicative inverse of n1 modulo n2:

n1^(-1) ≡ 1 (mod 1249)

Step 4: Solve the Congruences

We solve the congruences:

x ≡ 5 * n1^(-1) (mod n1n2) x ≡ 7 * n2^(-1) (mod n1n2)

Step 5: Find the Greatest Number

We find the greatest number that satisfies the congruences:

x ≡ 5 * n1^(-1) + 7 * n2^(-1) (mod n1n2)

Conclusion

In this article, we discussed how to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively. We used the Euclidean algorithm to find the greatest common divisor (GCD) of 281 and 1249, and then used the Chinese Remainder Theorem to find the greatest number that satisfies the congruences.

Final Answer

The final answer is 23.

Discussion

The greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively is 23. This is because 23 is the greatest common divisor (GCD) of 281 and 1249, and it satisfies the congruences:

x ≡ 5 (mod 281) x ≡ 7 (mod 1249)

References

  • Euclidean algorithm
  • Chinese Remainder Theorem

Tags

  • Greatest common divisor (GCD)
  • Euclidean algorithm
  • Chinese Remainder Theorem
  • Congruences
  • Modular multiplicative inverse

Introduction

In our previous article, we discussed how to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively. We used the Euclidean algorithm to find the greatest common divisor (GCD) of 281 and 1249, and then used the Chinese Remainder Theorem to find the greatest number that satisfies the congruences. In this article, we will answer some of the most frequently asked questions about this problem.

Q&A

Q: What is the greatest common divisor (GCD) of 281 and 1249?

A: The greatest common divisor (GCD) of 281 and 1249 is 1.

Q: Why is the GCD of 281 and 1249 1?

A: The GCD of 281 and 1249 is 1 because they are coprime numbers. This means that they have no common factors other than 1.

Q: What is the Chinese Remainder Theorem?

A: The Chinese Remainder Theorem is a mathematical theorem that states that if we have a system of congruences:

x ≡ a1 (mod n1) x ≡ a2 (mod n2)

where n1 and n2 are coprime, then there is a unique solution modulo n1n2.

Q: How do we use the Chinese Remainder Theorem to find the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively?

A: We use the Chinese Remainder Theorem by defining the congruences:

x ≡ 5 (mod 281) x ≡ 7 (mod 1249)

We then find the product of the moduli:

n1n2 = 281 * 1249

We find the modular multiplicative inverse of n1 modulo n2:

n1^(-1) ≡ 1 (mod 1249)

We solve the congruences:

x ≡ 5 * n1^(-1) (mod n1n2) x ≡ 7 * n2^(-1) (mod n1n2)

We find the greatest number that satisfies the congruences:

x ≡ 5 * n1^(-1) + 7 * n2^(-1) (mod n1n2)

Q: What is the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively?

A: The greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively is 23.

Q: Why is 23 the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively?

A: 23 is the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively because it is the greatest common divisor (GCD) of 281 and 1249, and it satisfies the congruences:

x ≡ 5 (mod 281) x ≡ 7 (mod 1249)

Conclusion

In this article, we answered some of the most frequently asked questions about the greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively. We used the Euclidean algorithm to find the greatest common divisor (GCD) of 281 and 1249, and then used the Chinese Remainder Theorem to find the greatest number that satisfies the congruences.

Final Answer

The final answer is 23.

Discussion

The greatest number that divides 281 and 1249, leaving a remainder of five and seven respectively is 23. This is because 23 is the greatest common divisor (GCD) of 281 and 1249, and it satisfies the congruences:

x ≡ 5 (mod 281) x ≡ 7 (mod 1249)

References

  • Euclidean algorithm
  • Chinese Remainder Theorem

Tags

  • Greatest common divisor (GCD)
  • Euclidean algorithm
  • Chinese Remainder Theorem
  • Congruences
  • Modular multiplicative inverse