Factoring and Multiples

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To determine which set of ratios are equivalent, we can simplify each pair of numbers. The ratio of 36 to 918 simplifies to 1:25.5, while 47 to 48 simplifies to approximately 0.979. The ratio of 12 to 34 is approximately 0.353, and 216 to 116 simplifies to approximately 1.862. None of the ratios are equivalent to each other.

The multiples of 4 up to 100 are the numbers obtained by multiplying 4 by the integers from 1 to 25. This results in the sequence: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, and 100. Thus, the last multiple of 4 before reaching 100 is 100 itself.

The sequence 5, 10, 20, 40, 80 can be identified as a geometric progression where each term is multiplied by 2. The nth term can be expressed as ( a_n = 5 \times 2^{(n-1)} ), where ( a_n ) is the nth term. Thus, for any integer ( n ), you can find the term by substituting ( n ) into this formula. For example, the 1st term is 5, the 2nd term is 10, and so on.

The number of factors of a given number corresponds to the different ways that number can be expressed as a product of two integers, which represents the possible dimensions of rectangular arrays. For instance, if a number has six factors, it can be arranged into rectangular arrays of dimensions that multiply to that number, such as 1x6, 2x3, and 3x2. Each unique pair of factors gives a distinct arrangement, illustrating the relationship between factors and rectangular arrays. Thus, the total number of factors directly determines the number of unique rectangular configurations possible for that number.

18 = 6 x 3

24 = 6 x 4

54 = 6 x 9

Since '6' appears in all multiples then '6' is the GCF.

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MDCCCCXX is the Roman numeral representation for the number 1920. In this notation, M represents 1000, D represents 500, C represents 100 (with four Cs making 400), and XX represents 20. Therefore, when combined, they total 1920.

No, 118 is not a multiple of 9. To determine if a number is a multiple of 9, you can sum its digits: 1 + 1 + 8 = 10. Since 10 is not divisible by 9, 118 is not a multiple of 9.

To find the largest perfect cube factor of 189, we first need to factor it into its prime components. The prime factorization of 189 is (3^3 \times 7^1). The largest perfect cube that can be formed from these factors is (3^3), which equals 27. Therefore, the largest perfect cube factor of 189 is 27.

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) ).

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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