To solve, write out every possible three-digit integer: 345, 354, 435, 453, 534, and 543; six different positive three-digit numbers can be formed, answer choice F.

Logarithms are used to indicate exponents of certain numbers called bases. By definition, log_a b = c, if a^c = b. To solve, let log₄ 64 = x; therefore, 64 = 4^x. Because 4³ = 64, x = 3.

To solve this problem, it might be helpful to use test values for r and s and systematically try the scenario presented in each answer choice. The equation |r − s|=|r + s| is true only when r = 0 or s = 0.

The sum of the interior angles of a pentagon is (5 − 2)(180◦), or 540◦. Thus, the total of the other 4 angles is 540◦ − 40◦, or 500◦.

A triangle with sides 9, 12, and 15 centimeters long has sides in the ratio of 9:12:15, which simplifies to 3:4:5. Recall that a 3 − 4 − 5 triangle is a special case because it is known to be a right triangle. Any triangle with sides in the same ratio is also a right triangle. Thus, there is a right angle between the smaller sides.

To find the distance between 2 points in the standard (x, y) coordinate plane you can use the distance formula d =\(\sqrt{[(x_{2} − x_{1})^{2} + (y_{2} − y_{1})^{2}]}\). Calculate the distance as follows:
\(\sqrt{[(4 − 1)^{2} + (4 − 0)^{2}]}=\sqrt{(3^{2}+4^{2})}=\sqrt{25}=5\)

To solve this problem, make x the smallest possible integer for which 15% of x is greater than 2.3. Then, set up the following inequality: 0.15x > 2.3. Divide both sides by 0.15 to get x > 15.333, repeating. The smallest integer greater than the repeating decimal 15.333 is 16.

You could take a “bruteforce” approach and test all the given values of y and see if you could find an x that worked. For example, if y = 9, then the two numbers are x²×9² and x × 93. You can see that 92 is a factor of these 2 numbers, so 27 cannot be the greatest common factor.
It might be more efficient to be more general and avoid testing all 5 values of x. Notice that xy² is a common factor of both x²y² and xy³. Because it is a factor, xy² must also be a factor of 27. Well, 27 factors as 3 × 32, so it seems natural to see if x = 3 and y = 3 are possible solutions. In this case the two numbers from the problem are 3² × 3² and 3³ × 3 and the greatest common factor is 3 × 3² = 27, so it works.

Systems of equations have an infinite number of solutions when the equations are equivalent. In order for the two equations to be equivalent, the constants and coefficients must be proportional. If the entire equation 3a + b = 12 is multiplied by 4, the result is 4(3a + b) = 4(12), or 12a + 4b = 48. Thus in order for the two equations to be equivalent, 3n = 48, or n = 16.

To solve this problem, first eliminate answer choices that yield equal values for f (−5) and f (5). These include answer choices in which the functions have even powers of x such as answer choice A, where f (x) = 6x², answer choice B, where f (x) = 6, and answer choice E, where f (x) = x⁶ + 6. Now, substitute −5 and 5 into the remaining answer choices:
Answer choice C: f (x) = 6/x. When x is −5, f (x) = −6/5, and when x is 5, f (x) = 6/5. Therefore, f (5) is greater than f (−5) and answer choice C is incorrect.
Answer choice D: f (x) = 6 − x³. When x = −5, f (x) = 6 − (−125) or 131, and when x is 5, f (x) = 6 − 125, or −116. Therefore, f (−5) is greater than f (5).