Find the Largest number which on Dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively

Find the Largest number which on Dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively

To find the largest number which, when dividing 1251, 9377, and 15628, leaves remainders of 1, 2, and 3 respectively, use the concept of the Highest Common Factor (HCF).

Subtract the given remainders from the numbers

1251 − 1 = 1250

9377 − 2 = 9375

15628 − 3 = 15625

Use Euclid's Division Lemma:

Use Euclid's division lemma to find the HCF of the numbers.

To find the HCF of 1250 and 9375:

9375 = 1250×7+625

Since the remainder is not zero, continue with 1250 and 625:

1250 = 625×2+0

The remainder is zero, so the HCF of 1250 and 9375 is 625.

To find the HCF of 15625 and 625:

15625 = 625×25+0

The remainder is zero, so the HCF of 15625 and 625 is 625.

The HCF of 1250, 9375, and 15625 is 625.

Therefore, the largest number that divides 1251, 9377, and 15628 leaving remainders of 1, 2, and 3 respectively is 625.

Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is a fundamental concept in mathematics that represents the largest number that can exactly divide two or more numbers without leaving a remainder. This concept is widely used in various mathematical problems and applications, making it essential to understand how to find the HCF efficiently.

Key Points

• The HCF of two or more numbers is the greatest number that divides each of them exactly.
• It is a measure of the common factors shared by the numbers.
• Understanding the HCF is crucial for simplifying fractions, solving problems involving divisibility, and finding common denominators.
• It helps in reducing fractions to their simplest form by dividing the numerator and the denominator by their HCF.

Methods to Find HCF:

Prime Factorization:

• This method involves breaking down each number into its prime factors and then identifying the common prime factors.
• The product of these common prime factors gives the HCF.

Euclidean Algorithm:

• This method uses the division process repeatedly to find the HCF.
• It involves dividing the larger number by the smaller number and taking the remainder, then repeating the process with the smaller number and the remainder until the remainder is zero.

Listing Factors:

Listing all the factors of each number and then identifying the largest common factor among them.

Steps to Find HCF:

• Identify the numbers for which the HCF needs to be found.
• Choose a method (Prime Factorization, Euclidean Algorithm, or Listing Factors).
• Apply the method systematically to find the HCF.
• Verify the HCF by checking that it divides each of the original numbers exactly.