Prime Numbers

2, 4, 8, 16, 32, 64, 128, 256, 512

858 is a composite numbers - its positive integer factors are:

1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858

Both composite.

2x5x5x3

you can apply the negative to any of the numbers

In drawing a plot line, there is an implicit assumption that a linear relationship hold over the range - although not necessarily outside it. Otherwise you could not draw any line - you could only mark the observed values. So, once you accept that assumption, it is valid to draw the line within the range.

100: 1, 2, 4, 5, 10, 20, 25, 50, 100
101: 1, 101
102: 1, 2, 3, 6, 17, 34, 51, 102
103: 1, 103
104: 1, 2, 4, 8, 13, 26, 52, 104
105: 1, 3, 5, 7, 15, 21, 35, 105
106: 1, 2, 53, 106
107: 1, 107
108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
109: 1, 109
110: 1, 2, 5, 10, 11, 22, 55, 110
111: 1, 3, 37, 111
112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
113: 1, 113
114: 1, 2, 3, 6, 19, 38, 57, 114
115: 1, 5, 23, 115
116: 1, 2, 4, 29, 58, 116
117: 1, 3, 9, 13, 39, 117
118: 1, 2, 59, 118
119: 1, 7, 17, 119
120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
121: 1, 11, 121
122: 1, 2, 61, 122
123: 1, 3, 41, 123
124: 1, 2, 4, 31, 62, 124
125: 1, 5, 25, 125
126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
127: 1, 127
128: 1, 2, 4, 8, 16, 32, 64, 128
129: 1, 3, 43, 129
130: 1, 2, 5, 10, 13, 26, 65, 130
131: 1, 131
132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
133: 1, 7, 19, 133
134: 1, 2, 67, 134
135: 1, 3, 5, 9, 15, 27, 45, 135
136: 1, 2, 4, 8, 17, 34, 68, 136
137: 1, 137
138: 1, 2, 3, 6, 23, 46, 69, 138
139: 1, 139
140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140
141: 1, 3, 47, 141
142: 1, 2, 71, 142
143: 1, 11, 13, 143
144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
145: 1, 5, 29, 145
146: 1, 2, 73, 146
147: 1, 3, 7, 21, 49, 147
148: 1, 2, 4, 37, 74, 148
149: 1, 149
150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

No, it is not a prime number.

No, it is not a prime number.

It has more than two factors.

A prime number is a number with two and only two factors: 1 (also known as unity) and itself. A composite number is a number with more than two factors; in other words, it has at least one factor besides 1 and itself. 556 has factors other than 1 and itself, so it is not a prime number. For example, it is divisible by 2 because it is an even number. Therefore, 556 a composite number.

In addition to ' 1 ' and itself, 18 has the additional factors 2, 3, 6, and 9,

so it's not prime.

Composite means it would have more than 1 and itself as factors, which it does; however, if it's prime, the only two factors are 1 and the number given.

1

2 x 2 x 2 x 2 = 16

2 x 2 x 3 = 12

Yes.

Integers greater than 1 can be prime or composite. No integer can be prime composite.

The factors of 160 are:

1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160

The sum of the first six prime numbers is 41 (2 + 3 + 5 + 7 + 11 + 13 = 41).

29 and 47 are prime.

No other prime numbers are consecutive because there aren't any other even prime numbers.

There are several advanced methods, but they are all much more complicated than checking for factors. They are, however, efficient for extremely large numbers (such as the 300-digit or so numbers commonly used in cryptography).If you are actually interested in these (more complicated) methods, check the Wikipedia, or do a Google search, on the topic "primality tests".

It is a composite.

201 is a composite number because it has more than two factors whereas a prime number has only two factors

+/- sqrt(61).

The factors are of the form (ax + b)(cx +d)(ex + f) then, the coefficient of x³ is the product of the coefficients of the three x terms, namely ace = 6

The coefficient of x² is the sum of the product of the coefficients of x from 2 brackets multiplied by the number from the remaining bracket.

The coefficient of x is the sum of the product of the numbers from two brackets multiplied by the x coefficient from the remaining bracket.

The final number is the product of the numbers from the three brackets giving bdf = 4.

Some number crunching gives the solution (3x - 1)(x - 4)(2x + 1)

Alternatively:

We are told that 2x + 1 is a factor. Thus we know:

(2x + 1)(ax² + bx + c) = 6x³ - 23x² - 5x + 4

Multiplying out the left gives:

2ax³ + (2b + a)x² + (2c + b)x + c = 6x³ - 23x² - 5x + 4

Thus: 2a = 6 (coefficients of x³) → a = 3

And 2b + a = 2b + 3 = -23 (coefficients of x²) → b = -13

And c = 4 (term not including any x).

Thus we now know;

6x³ - 23x² - 5x + 4 = (2x +1)(3x² - 13x + 4)

As we are told to factorise it completely, it is likely that the quadratic also factorises. As 3 only has the factor pair 1×3, the quadratic will factorise into (3x + d)(x + e)

Thus 3x³ + (3e + d)x + de = 3x² -13x + 4

As the term without any x is positive, d and e must have the same sign.

As the coefficient of the x term is negative, both d and e must be negative.

Thus, the factor pairs of 4 to consider are (-1)×(-4) and (-2)×(-2).

As the coefficient of x is -13 = 3e + d, the values e = -4 and -1 are [fairly] obvious.

Thus 6x³ - 23x² - 5x + 4 = (2x +1)(3x - 1)(x - 4)

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