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Chandu M.

Because everyone, even the smartest and brightest, needs help sometimes, I want to offer my experiences and advice whenever possible.

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Number Theory

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Question:

If both 112 and 33 are factors of the number a * 43 * 62 * 1311, what is the smallest possible value of 'a'?

Chandu M.

Answer:

112 is a factor of the given number. In the given expression, a * 43 * 62 * 1311 none of the other factors, viz., 4, 6 or 13 is either a power or multiple of 11. Hence, if a * 43 * 62 * 1311 is divisible by 112, 'a' should necessarily include 112. The question states that 33 is a factor of the given number. 62 is a part of the number. 62 can be expressed as 32 * 22. i.e., a * 43 * 62 * 1311 has 32 in it. It needs a 33 in it for the number to be divisible by 33. Therefore, a will have to provide one more 3 to a * 43 * 62 * 1311. Therefore, 'a' should be at least 112 * 3 = 363 if the given number has to have 112 and 33 as its factors.

Statistics

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Question:

In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state. What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

Chandu M.

Answer:

Make sure the sample size is big enough to model differences with a normal population. Because n1P1 = 100 * 0.52 = 52, n1(1 - P1) = 100 * 0.48 = 48, n2P2 = 100 * 0.47 = 47, and n2(1 - P2) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough. Find the mean of the difference in sample proportions: E(p1 - p2) = P1 - P2 = 0.52 - 0.47 = 0.05. Find the standard deviation of the difference. σd = sqrt{ [ P1(1 - P1) / n1 ] + [ P2(1 - P2) / n2 ] } σd = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] } σd = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706 Find the probability. This problem requires us to find the probability that p1 is less than p2. This is equivalent to finding the probability that p1 - p2 is less than zero. To find this probability, we need to transform the random variable (p1 - p2) into a z-score. That transformation appears below. z p1 - p2 = (x - μ p1 - p2 ) / σd = = (0 - 0.05)/0.0706 = -0.7082 Using Stat Trek's Normal Distribution Calculator, we find that the probability of a z-score being -0.7082 or less is 0.24.

Algebra

TutorMe

Question:

Write 230,000,000,000 in scientific notation.

Chandu M.

Answer:

a × 10 n , where a is a real number such that 1 ≤ |a| < 10 and n is an integer. 230,000,000,000 = 2.3 × 100,000,000,000 = 2.3 × 10 11

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