Find the hcf of the three numbers 442,569,697.I,e of 442-1,569-2,and 697-4.The required number is63

Problem: Find the greatest number that divides 442,569,697 leaving remainders 1, 2, and 4 respectively.

Solution:

To find the greatest number that satisfies the given condition, we need to find the common divisor of the differences between the given number and the remainders.

Step 1: Find the differences
Let's subtract the remainders from the given number:
- Difference for remainder 1: 442,569,697 - 1 = 442,569,696
- Difference for remainder 2: 442,569,697 - 2 = 442,569,695
- Difference for remainder 4: 442,569,697 - 4 = 442,569,693

Step 2: Find the common divisor
Now, we need to find the greatest common divisor (GCD) of the differences calculated in step 1. GCD is the largest positive integer that divides all the given numbers.

To find the GCD, we can use the Euclidean algorithm. Let's calculate the GCD of the differences:

- GCD(442,569,696, 442,569,695) = GCD(442,569,696, 1)
- GCD(442,569,696, 1) = GCD(1, 442,569,696 % 1) = GCD(1, 0) = 1

Therefore, the greatest number that divides 442,569,697 leaving remainders 1, 2, and 4 respectively is 1.

Explanation:

To understand why 1 is the greatest number that satisfies the given condition, let's consider the properties of remainders and divisors.

1. Remainders: When dividing a number by any positive integer, the remainder can range from 0 to (divisor - 1). In this case, the remainders are 1, 2, and 4, which means the divisor must divide the given number and leave one of these remainders.

2. Divisors: A divisor is a positive integer that divides another integer without leaving a remainder. The greatest divisor of any number is the number itself. In this case, the given number is 442,569,697, and it is divisible by itself.

Since all positive integers are divisible by 1 without leaving a remainder, 1 satisfies the given condition. Therefore, the greatest number that divides 442,569,697 leaving remainders 1, 2, and 4 respectively is 1.

Summary:

The greatest number that divides 442,569,697 leaving remainders 1, 2, and 4 respectively is 1. This is because all positive integers are divisible by 1 without leaving a remainder.