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3 years ago

Find the highest common factor and lowest common multiple of 60 and 150 using prime factors

1 answer:

7 0

Here, we know, 60 = 2 × 30
& 150 = 5 × 30

So, Highest Common Factor (HCF) would be 30

Now, Lowest common point = 60 × 5 = 300 or 150 × 2 = 300

In short, LCM would be 300

Hope this helps!

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Problems 2.21, 2.22, 2.23

RoseWind [281]

Statements can be proved by contrapositive, contradiction or by induction.

2.21 and 2.23 are proved by contrapositive
2.22 is proved by induction

2.21: If $n^3$ is even, then n is even (By contrapositive)

The contrapositive of the above statement is that:

If n is odd, then $n^3$ is odd

Represent the value of n as:

$n = 2k + 1$ , where $k \ge 0$

Take the cube of both sides

$n^3 = (2k + 1)^3$

Expand

$n^3 = 8k^3 + 6k^2 + 6k + 1$

Group

$n^3 =[ 8k^3 + 6k^2 + 6k] + 1$

Factor out 2

$n^3 =2[4k^3 + 3k^2 + 3k] + 1$

Assume w is an integer; where:

$w =4k^3 + 3k^2 + 3k$

So, we have:

$n^3 =2w + 1$

The constant term (i.e. 1) means that $n^3$ is odd.

Hence, the statement has been proved by contrapositive.

i.e. If n is odd, then $n^3$ is odd

2.22 $3n + 4$ is even, if and only if n is even

We have: $3n + 4$

Assume that: $n = 2k + 2$ for $k \ge 0$

So, we have:

$3n + 4 = 3(2k + 2) + 4$

Open bracket

$3n + 4 = 6k + 6 + 4$

$3n + 4 = 6k + 10$

Factorize

$3n + 4 = 2(3k + 5)$

The factor of 2 means that $3n + 4$ is even.

Hence, $3n + 4$ is even, if and only if n is even

2.22: $s \ne -1$ and $t \ne -1$ , then $s + t + st \ne -1$

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

We have:

$s + t + st = -1$

Substitute the values of s and t in: $s + t + st = -1$

$-1 -1 -1 \times -1 = -1$

$-1 -1 + 1 = -1$

$-1 = -1$

Hence, by contrapositive:

If $s = -1$ and $t=-1$ , then $s + t + st = -1$

Read more about proofs at:

brainly.com/question/19643658

7 0

3 years ago

The average cost of living for a family of four in twelve different cities was found to be $65,351 with a standard deviation of

Andrej [43]

Answer:

$65,351+/-$3,661.73

= ( $61,689.27, $69,012.73)

Therefore, the 90% confidence interval (a,b) = ( $61,689.27, $69,012.73)

Step-by-step explanation:

Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.

The confidence interval of a statistical data can be written as.

x+/-zr/√n

Given that;

Mean x = $65,351

Standard deviation r = $7,711

Number of samples n = 12

Confidence interval = 90%

z(at 90% confidence) = 1.645

Substituting the values we have;

$65,351+/-1.645($7,711/√12)

$65,351+/-1.645($2,225.973962860)

$65,351+/-$3661.727168905

$65,351+/-$3,661.73

= ( $61,689.27, $69,012.73)

Therefore, the 90% confidence interval (a,b) = ( $61,689.27, $69,012.73)

6 0

3 years ago

According to the Complex Conjugate Root Theorem, if a+bi is a root of a quadratic equation, then __blank__ is also a root of the

Trava [24]

Answer:

a-bi

Step-by-step explanation:

If a quadratic equation lx^2+mx+n=0 has one imaginary root as a+bi then the other root is the conjugate of a+bi = a-bi

Because we have l, m and n are real numbers and they are the coefficients.

Sum of roots = a+bi + second root = -m/l

When -m/l is real because the ratio of two real numbers, left side also has to be real.

Since bi is one imaginary term already there other root should have -bi in it so that the sum becomes real.

i.e. other root will be of the form c-bi for some real c.

Now product of roots = (a+bi)(c-bi) = n/l

Since right side is real, left side also must be real.

i.e.imaginary part =0

bi(a-c) =0

Or a =c

i.e. other root c-bi = a-bi

Hence proved.

3 0

3 years ago

Can someone tell me this? I dont get it.

Harman [31]

Answer:

Step-by-step explanation:

Numbers 1 and 4 are perimeter questions so all you do is add up all the sides and the label is cm. For #2, this is an area problem and the area for a rectangle is length times width, so multiply the 2 together and the label is inches squared. For #3, the volume formula for a cube is side*side*side and the label for this is feet cubed. For #5, the area for a triangle is .5 times the base times the height, and the label is mm-squared. That should help you get all the answers on your wn. Very, very simple.

3 0

3 years ago

X.7=14 plz help me 4th grade work

OlgaM077 [116]

x*7=14

14/7=2

x=2

brainliest please

8 0

3 years ago