The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, except for the order of the factors. In other words:

• Every positive integer greater than 1 can be factored into a product of prime numbers.
• This factorization is unique up to the order of the factors.

For example, let's take the number 12:

12=2×2×312=2×2×3

This is one possible prime factorization of 12. According to the Fundamental Theorem of Arithmetic, any other factorization of 12 into prime numbers will involve the same prime factors (in this case, 2 and 3), just in a different order.

This theorem is fundamental in number theory and is essential for many mathematical proofs and concepts, including the understanding of factors, divisors, and properties of numbers.

To draw a line segment of length 12.8 cm and divide it into four equal parts using compasses, follow these steps:

1. Draw the Line Segment:

   • Use a ruler and a pencil to draw a straight line segment of any convenient length. Label the endpoints of the segment as A and B.

2. Measure and Mark the Length:

   • Starting from point A or B, measure and mark a distance of 12.8 cm along the line segment. Label this point as C. This will ensure that the line segment is exactly 12.8 cm long.

3. Divide the Line Segment into Four Equal Parts:

   • With C as the center, use a compass to draw an arc above and below the line segment. The radius of the arcs should be greater than one-fourth of the length of AB.
   • Without changing the compass width, draw two more arcs with the same radius, each centered at the previous intersection points. Repeat this process until you have four intersection points equally spaced along the line segment.
   • Use a ruler to draw straight lines connecting the adjacent intersection points. These lines will divide the line segment AB into four equal parts.

4. Verify by Actual Measurement:

   • Use a ruler to measure each of the four parts. If they are indeed equal, then the division is accurate.

By following these steps carefully, you can divide the line segment of length 12.8 cm into four equal parts using compasses. Make sure to verify the division by actual measurement to ensure accuracy.

To write the given numbers in ascending order:

1. 63,854
2. 63,584
3. 65,348
4. 64,835
5. 68,543

To determine how many of these numbers are even, we look at the units place of each number. If the units place is an even digit (0, 2, 4, 6, 8), then the number is even.

Looking at the units place of each number:

• 63,854: Even
• 63,584: Even
• 65,348: Even
• 64,835: Odd
• 68,543: Odd

Out of the given numbers, 3 of them are even.

To find the ratio of 75 cm to 1.5 m, we need to convert both quantities to the same unit of measurement.

Since 1 meter (m) is equal to 100 centimeters (cm), we can convert 1.5 m to centimeters: 1.5 m×100 cm/m=150 cm1.5m×100cm/m=150cm

Now, we have:

• 75 cm
• 150 cm

To find the ratio, we divide the first quantity by the second: Ratio=75 cm150 cmRatio=150cm75cm

Ratio=75150Ratio=15075

Ratio=0.5Ratio=0.5

So, the ratio of 75 cm to 1.5 m is 0.5.