Factoring and Multiples

2)48

2)24

2)12

2)6

3)3 =1

So factors are 2x2x2x2x3 = 2^(4) x 3^(1)

2 and 3 are factors of 48 and prime numbers.

Prime numbers have exactly two factors.
All prime numbers have only 2 factors whereas composite numbers have more than 2 factors

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We note that 8 = 4 x 2'

So 4 x 7 = 28

So mulktply 28 by '2'

Hence 28 x 2 = 56 The answer!!!!!

To convert 49.7 kilograms to stones, you divide by 6.35 (since 1 stone is approximately 6.35 kg). This results in approximately 7.83 stones. Therefore, 49.7 kg is roughly equivalent to 7 stone and 11 pounds.

The sequence -1, -8, -27, -64 consists of negative perfect cubes: specifically, -1 is (-1^3), -8 is (-2^3), -27 is (-3^3), and -64 is (-4^3). The conjecture suggests that the next term in the sequence would be (-125), which is (-5^3). Thus, the pattern follows the form of (-n^3) for (n = 1, 2, 3, 4, ...).

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To write ( x(x-2) + 3(2-x) ) as a product of binomials, first simplify the expression. Notice that ( 2 - x ) can be rewritten as ( -(x - 2) ), so ( 3(2-x) = -3(x-2) ). Combining the terms, we have ( x(x-2) - 3(x-2) = (x-2)(x-3) ). Thus, the expression can be factored as ( (x-2)(x-3) ).

The formula for the circumference of a circle is given by ( C = 2 \pi r ), where ( \pi ) is approximately 3.14 and ( r ) is the radius of the circle. To find the circumference, simply multiply 2, 3.14, and the radius ( r ). For example, if the radius is 5, the circumference would be ( C = 2 \times 3.14 \times 5 = 31.4 ).

191 is prime. It has one pair of factors, one and itself.

Mixed Numbers/Fractions.

Here is an example of multiplication.

2 2/5 X 3 4/7

First convert to improper(top heavy) fractions.

12/5 X 25/7

Cancel down(reduce) if possible. In this case by '5'.

Hence

12/1 X 5/7

Multiply the numerators(top) numbers together. Then multiply the denominators(bottom) numbers together.

Hence

60/7

Convert back to a mixed number/fraction, by dividing '60' by '7.

60/7 = 8 4/7 ( The answer).

Here is an example of Division

4 1/ 7 divide 3 4/5

Again convert to improper fractions.

29/7 divide 19/5

This next bit is very important. 'CHANGE THE SIGN TO MULTIPLY AND INVERT(TURN OVER) THE RIGHT HAND FRACTION.' Then proceed as multiplication.

Hence

29/7 X 5/19

Cancel down if appropriate. We cannot cancel down in this case , as there are no common factors 'top and bottom'.

Multiply the numerators(top) numbers together. Then multiply the denominators(bottom) numbers together.

Hence

145/133

Reduce to a mixed number/fraction

1 12/133 The Answer!!!!!

NB I have deliberately selected 'big' numbers to show how to manipulate large numbers in fractions.

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The phrase "the greatest good for the greatest number" is commonly associated with the utilitarian philosophy of Jeremy Bentham. He is known for advocating that the moral worth of an action is determined by its contribution to overall happiness or utility. This principle emphasizes maximizing well-being and minimizing suffering for the largest number of people.

The factors of 696 are 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, and 696. The prime factors of 696 are 2 x 2 x 2 x 3 x 29.

The highest point on Interstate 40 is located near the New Mexico and Arizona border, specifically at the Continental Divide in the vicinity of the town of Alamogordo, New Mexico. This elevation reaches approximately 7,300 feet (2,225 meters) above sea level. The scenery around this area features stunning vistas of the surrounding landscape, including the Rocky Mountains and the high desert terrain.

To find factors of 17424 that add up to -264, we can consider pairs of factors whose product is 17424. One such pair is -288 and 24, since -288 + 24 = -264. Thus, the relevant factors of 17424 that meet the criteria are -288 and 24.

The sequence appears to follow the pattern of ( n^3 - 1 ) for ( n = 1, 2, 3, 4, 5 ). Specifically, the numbers correspond to ( 1^3 - 1 = 0 ), ( 2^3 - 1 = 7 ), ( 3^3 - 1 = 26 ), and ( 4^3 - 1 = 63 ). Therefore, the next number would be ( 5^3 - 1 = 124 ). Thus, the next number in the sequence is 124.

The numbers below are the factor pairs of 290. Multiplied together they equal 290:

(1, 290)

(2, 145)

(5, 58)

(10, 29)

The pattern in the sequence 0, 7, 26, 63, 124 involves the formula ( n^3 - n ), where ( n ) is a positive integer starting from 1. Specifically, for each ( n ):

• ( 1^3 - 1 = 0 )
• ( 2^3 - 2 = 7 )
• ( 3^3 - 3 = 26 )
• ( 4^3 - 4 = 63 )
• ( 5^3 - 5 = 124 )

Thus, the next number in the sequence would be ( 6^3 - 6 = 210 ).

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In the context of mathematics, the numbers 7 and 49 have specific factors due to their properties. The number 7 is a prime number, meaning its only factors are 1 and 7 itself. On the other hand, 49 is a composite number that can be expressed as 7 multiplied by itself (7 x 7), so its factors include 1, 7, and 49. Thus, the only shared factors between the two numbers are 1 and 7, leading to the conclusion that these are the only facts in their "family."

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