Find [tex]\operatorname{gcd}(1000,625)[/tex] and [tex]\operatorname{lcm}(1000,625)[/tex] and verify that [tex]\operatorname{gcd}(1000,625) \cdot \operatorname{lcm}(1000,625)=1000 \cdot 625[/tex].



Answer :

Sure! Let's find [tex]\(\operatorname{gcd}(1000, 625)\)[/tex] and [tex]\(\operatorname{lcm}(1000, 625)\)[/tex], and verify the property that [tex]\(\operatorname{gcd}(1000, 625) \cdot \operatorname{lcm}(1000, 625) = 1000 \cdot 625\)[/tex].

### Step 1: Finding the [tex]\(\operatorname{gcd}(1000, 625)\)[/tex]

The greatest common divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder.

Given:
- [tex]\(a = 1000\)[/tex]
- [tex]\(b = 625\)[/tex]

Using the Euclidean algorithm, we can find the gcd of 1000 and 625.

1. Divide 1000 by 625 and find the remainder:
[tex]\[ 1000 = 625 \times 1 + 375 \][/tex]
Remainder is 375.

2. Now, divide 625 by 375:
[tex]\[ 625 = 375 \times 1 + 250 \][/tex]
Remainder is 250.

3. Next, divide 375 by 250:
[tex]\[ 375 = 250 \times 1 + 125 \][/tex]
Remainder is 125.

4. Finally, divide 250 by 125:
[tex]\[ 250 = 125 \times 2 + 0 \][/tex]
Remainder is 0.

Since the remainder is now 0, the gcd is the last non-zero remainder, which is:
[tex]\[ \operatorname{gcd}(1000, 625) = 125 \][/tex]

### Step 2: Finding the [tex]\(\operatorname{lcm}(1000, 625)\)[/tex]

The least common multiple (lcm) of two numbers is the smallest number that is a multiple of both. The relationship between gcd and lcm of two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is given by:
[tex]\[ \operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b) = a \cdot b \][/tex]

We know:
[tex]\[ \operatorname{gcd}(1000, 625) = 125 \][/tex]
[tex]\[ a \cdot b = 1000 \cdot 625 = 625000 \][/tex]

Thus, we can find the lcm using:
[tex]\[ \operatorname{lcm}(1000, 625) = \frac{1000 \cdot 625}{\operatorname{gcd}(1000, 625)} = \frac{625000}{125} = 5000 \][/tex]

So, the lcm is:
[tex]\[ \operatorname{lcm}(1000, 625) = 5000 \][/tex]

### Step 3: Verifying the Property

To confirm the relationship:
[tex]\[ \operatorname{gcd}(1000, 625) \cdot \operatorname{lcm}(1000, 625) = 1000 \cdot 625 \][/tex]

Substitute the obtained values:
[tex]\[ 125 \cdot 5000 = 625000 \][/tex]
[tex]\[ 1000 \cdot 625 = 625000 \][/tex]

Both sides are equal. Thus, the property is verified.

### Conclusion

- [tex]\(\operatorname{gcd}(1000, 625) = 125\)[/tex]
- [tex]\(\operatorname{lcm}(1000, 625) = 5000\)[/tex]
- The property [tex]\(\operatorname{gcd}(1000, 625) \cdot \operatorname{lcm}(1000, 625) = 1000 \cdot 625\)[/tex] holds true.