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Sagot :
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.
### 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.
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