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3 years ago

What is the last digit of 3^36th power? Please explain, I still do not understand why it is 0.

Mathematics

2 answers:

Dafna11 [192]3 years ago

3 0

The answer is actually 1, and not 0, and here's why.

Write 3^36 into (3^3)^12. Note that the exponents 3 and 12 multiply to 36

Now focus on 3^3 which turns into 27. The last digit here is 7. It turns out that this is the only thing that matters when it comes to finding the final answer.

So instead of 27^12, we can look at 7^12

Break 7^12 into (7^3)^4, then note how 7^3 = 343. Again the units digit is all we care about. So instead of 343^4, we just look at 3^4

3^4 = 81 and the units digit here is 1 which is the final answer.

---------------------------------

Here's an alternative approach using modular arithmetic. This is the set of tools that helps us find remainders. If we work with mod 10, then we can find the units digit of any complicated expression such as 3^36

3^36 = (3^3)^12 (mod 10)

3^36 = 27^12 (mod 10)

3^36 = 7^12 (mod 10)

3^36 = (7^3)^4 (mod 10)

3^36 = 343^4 (mod 10)

3^36 = 3^4 (mod 10)

3^36 = 81 (mod 10)

3^36 = 1 (mod 10)

We effectively follow the same steps as mentioned above, but we're using a more formal set of steps to show.

Schach [20]3 years ago

3 0

Answer:

The last digit is 1.

Step-by-step explanation:

3^36 = 150094635296999121

The last digit is 1.

The digit cannot be zero because there are no factors of 10 in 3^36.

_____

Consider the last digits of powers of 3:

3^0 = 1

3^1 = 3

3^2 = 9

3^3 mod 10 = 7

3^4 mod 10 = 3·7 mod 10 = 1

The last digit repeats in the pattern 1, 3, 9, 7, 1. So every 4th exponent will have the same result.

So, 3^36 mod 10 = 3^(36 mod 4) = 3^0 = 1.

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2 years ago

In a random sample of males, it was found that 28 write with their left hands and 210 do not. In a random sample of females, it

Drupady [299]

Answer:

a) Null hypothesis: $p_{M} \geq p_{W}$

Alternative hypothesis: $p_{M} < p_{W}$

b) $z=\frac{0.118-0.125}{\sqrt{0.122(1-0.122)(\frac{1}{238}+\frac{1}{497})}}=-0.271$

c) $p_v =P(Z$

d) If we compare the p value and the significance level given $\alpha=0.05$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion of men that write with their left hand is NOT significant lower than the proportion of female that write with their left hand

Step-by-step explanation:

1) Data given and notation

$X_{M}=28$ represent the number of men that write with their left hand

$X_{W}=62$ represent the number of women that write with their left hand

$n_{M}=210+28=238$ sample of male selected

$n_{W}=62+435=497$ sample of female selected

$p_{M}=\frac{28}{238}=0.118$ represent the proportion of men that write with their left hand

$p_{W}=\frac{62}{497}=0.125$ represent the proportion of women that write with their left hand

$\alpha=0.05$ represent the significance level

z would represent the statistic (variable of interest)

$p_v$ represent the value for the test (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to check if the rate of left-handedness among males is less than that among females , the system of hypothesis would be:

Null hypothesis: $p_{M} \geq p_{W}$

Alternative hypothesis: $p_{M} < p_{W}$

We need to apply a z test to compare proportions, and the statistic is given by:

$z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}$ (1)

Where $\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{28+62}{238+497}=0.122$

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:

$z=\frac{0.118-0.125}{\sqrt{0.122(1-0.122)(\frac{1}{238}+\frac{1}{497})}}=-0.271$

4) Statistical decision

We have a significance level provided $\alpha=0.05$ , and now we can calculate the p value for this test.

Since is a one left side test the p value would be:

$p_v =P(Z$

If we compare the p value and the significance level given $\alpha=0.05, 0,1,0.15$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion of men that write with their left hand is NOT significant loer than the proportion of female that write with their left hand .