Learn Maths with Free Lessons & Tips

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 simplify the expression 18+{1+(5−3)×5}18+{1+(5−3)×5}, we need to follow the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, let's simplify the expression inside the innermost parentheses:

5−3=25−3=2

Now, let's simplify the expression inside the curly braces:

1+2×5=1+10=111+2×5=1+10=11

Now, substitute the result back into the original expression:

18+1118+11

Now, we just need to perform the addition:

18+11=2918+11=29

So, the simplified expression is 2929.

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.

The smallest three-digit number whose value does not change when its digits are reversed is a palindrome. Palindromic numbers are the same when read forwards and backwards.

The smallest three-digit palindromic number is 101.

This number reads the same forwards and backwards, so its value does not change when its digits are reversed.

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.

Using a divider rather than a ruler to measure the length of a line segment can offer several advantages:

1. Precision: Dividers typically have sharper points than rulers, allowing for more precise placement on the endpoints of the line segment. This precision helps ensure that the measurement accurately reflects the length of the segment.

2. Avoiding Parallax Errors: When using a ruler, there's a risk of parallax errors, where the position of the observer's eye affects the perceived position of the markings on the ruler. Dividers eliminate this issue because their points can be placed directly on the endpoints of the segment without needing to align with markings.

3. Consistency: Dividers provide consistent results regardless of the scale or markings on the ruler. With a ruler, variations in the thickness of the markings or the alignment of the observer's eye can lead to inconsistent measurements.

4. Versatility: Dividers can be adjusted to match the length of the segment precisely without the need for additional markings or measurements. This versatility makes dividers suitable for measuring segments of any length or scale.

5. Minimizing Damage: Dividers exert less pressure on the paper or surface being measured compared to rulers, reducing the risk of causing indentations or damage to delicate materials.

Overall, while rulers have their uses, dividers are often preferred for precise and consistent measurements of line segments due to their sharper points, avoidance of parallax errors, versatility, and gentler treatment of materials.

To determine which point lies between the other two, we can compare the sum of the lengths of two line segments with the length of the third segment.

Let's compare the sum of AB and BC with AC:

AB + BC = 5 cm + 3 cm = 8 cm

Since AB + BC = AC (8 cm), this means that points A and C are consecutive and point B lies between them.

So, point B lies between points A and C.

Certainly! Here are five English letters that have a horizontal line of symmetry:

1. Letter "A"
2. Letter "H"
3. Letter "M"
4. Letter "O"
5. Letter "U"