In the number \(12ab\), \(a\) and \(b\) are digits. Find \(a\) and \(b\) such that \(12ab\) is divisible by 2, 5, and 7.
A) \(a = 2, b = 7\)
B) \(a = 2, b = 5\)
C) \(a = 7, b = 0\)
D) \(a = 7, b = 5\)
E) \(a = 6, b = 0\)
What is the average (arithmetic mean) of all multiples of 6 from 6 to 510 inclusive?
A) 516
B) 258
C) 1530
D) 252
E) 1510
Which of the following numbers can be expressed as the product of 3 different integers greater than 1?
I) 24
II) 27
III) 48
A) I and III only
B) I and II only
C) I, II, and III
D) I only
E) III only
If 60% of \(a\) is equal to 120% of \(b\), then
\[2a - 4b = ?\]A) \(2a\)
B) \(0.5a\)
C) \(0\)
D) \(3a\)
E) \(-360\)
\[ \frac{4^{1000} \cdot 8^{2000}}{16^{2000}} = ? \]
A) 2
B) 3
C) 4
D) 1
E) 5
If \(n\) is a positive integer divisible by 2, 5, and 7, which of the following is true?
I) \(n^2 + 100\) is divisible by 100
II) \(n + 28\) is divisible by 14
III) \(n^2 + 1\) is divisible by 2
A) I and II only
B) I, II, and III
C) None
D) I and III only
E) II and III only
A spherical block of metal weighs 12 pounds. What is the weight, in pounds, of another block of the same metal if its radius is 3 times the radius of the 12‑pound block?
A) 324
B) 15
C) 36
D) 108
E) 4
If the ages of students in a class vary between 18 and 22 inclusive, which of the following describes all possible ages \(x\) in this class?
A) \(|x - 18| \le 4\)
B) \(|x + 20| \le 2\)
C) \(|x - 21| \le 1\)
D) \(|x - 22| \le 0\)
E) \(|x - 20| \le 2\)
If \(r\%\) of \(y\) is \(A\), what is \(y\)?
A) \(\frac{A}{r}\)
B) \(\frac{A}{100r}\)
C) \(\frac{r}{100A}\)
D) \(\frac{100r}{A}\)
E) \(\frac{100A}{r}\)
Five years ago Ben was 4 times as old as Julie. If Julie is 10 years old now, how old is Ben?
A) 40
B) 25
C) 14
D) 19
E) 30
If \(\frac{x+2}{7}\) is an integer greater than 2, then the remainder when \(x\) is divided by 7 is
A) 2
B) 5
C) 6
D) 4
E) 3
For how many positive integer values of \(n\) will the expression \(2n^2 + 1\) be an integer greater than 2 and less than 400?
A) 16
B) 32
C) 28
D) 14
E) 15
If \(x + y = \sqrt{22}\) and \(x - y = \sqrt{10}\), then \(xy =\)
A) \(\sqrt{22} \sqrt{10}\)
B) 3
C) 12
D) 55
E) 16
In the figure below, the three circles have equal radii and are tangent to each other and to the sides of the rectangle. The width of the rectangle is 20 feet. What is the area of the shaded (red) region outside and enclosed by all three circles?
A) \(\frac{50\sqrt{3}}{2}\)
B) \(25\pi\)
C) \(2\sqrt{3} + 25\pi\)
D) \(50\sqrt{3} - 25\pi\)
E) \(\frac{25(2\sqrt{3} - \pi)}{2}\)
If \(2x + 1 = -x + 3\), what is the value of \(\frac{2}{3}x + \frac{1}{3}\)?
A) 1
B) \(\frac{1}{3}\)
C) \(\frac{2}{3}\)
D) \(\frac{7}{9}\)
E) \(\frac{5}{9}\)
If \(a < b < c < d\) and the average (arithmetic mean) of \(a, b, c, d\) is \(m\), which of the following is true?
I) \(a + b < c + d\)
II) \(m < d\) and \(a < m\)
III) \(b < m < c\)
A) II only
B) I only
C) III only
D) I and II only
E) II and III only
If \(x, y, z\) are positive numbers such that \(5x = \frac{y}{4}\), \(\frac{y}{4} = \frac{z}{5}\), and \(y + z = \frac{x}{k}\), what is the value of \(k\)?
A) \(\frac{1}{2}\)
B) \(\frac{1}{25}\)
C) \(\frac{1}{5}\)
D) \(\frac{1}{45}\)
E) \(\frac{5}{4}\)
Two square tables have sides of 10 and 15 inches respectively. The area of the larger table is what percent more than the area of the smaller table?
A) 50%
B) 15%
C) 125%
D) 25%
E) 5%
Find \(-a + b - c + d\) if
\[(x - \tfrac{1}{5})(x + 1)(x + \tfrac{3}{7}) = ax^3 + bx^2 + cx + d\]for all real \(x\).
A) 0
B) 1
C) 2
D) 3
E) 4
In the figure below, line \(l\) is parallel to line \(k\) and line \(g\) is parallel to line \(h\). What is the value of \(y\)?
A) 10
B) 20
C) 5
D) 15
E) 35
Line \(l\) has equation \(y - 2x = 2\). What is the equation of line \(p\) which is the reflection of line \(l\) across the line \(y = x\)?
A) \(y = 0.5x - 1\)
B) \(y = 2x - 1\)
C) \(y = x + 2\)
D) \(y = -2x - 2\)
E) \(y = x\)
Mike drove 30 miles at a constant speed for \(t\) hours and then drove \(y\) miles at another constant speed for 1 hour 15 minutes. What was his average speed, in miles per hour, for the whole journey?
A) \(\frac{\frac{30}{t} + \frac{y}{1.25}}{2}\)
B) \(\frac{30 + y}{t + 1.25}\)
C) \(\frac{30 + y}{t}\)
D) \(\frac{30 + y}{1.25}\)
E) \(\frac{30 + y}{2(t + 1.25)}\)
If \(M\) and \(N\) are negative integers and \(-3M + 4N = 10\), which of the following could be the value of \(N\)?
A) \(-2\)
B) \(-4\)
C) \(-6\)
D) \(-7\)
E) \(-9\)
If \(x^6 = 20\), what is \(x^8\)?
A) \(400\sqrt[3]{20}\)
B) \(10\sqrt{20}\)
C) \(20\sqrt{20}\)
D) \(20\sqrt[3]{20}\)
E) \(\sqrt[3]{20}\)
\(n\) is an integer chosen at random from \(\{2,5,6\}\) and \(p\) another integer chosen at random from \(\{6,9,10\}\). What is the probability that both \(n\) and \(p\) are even?
A) \(\frac{2}{9}\)
B) \(\frac{5}{9}\)
C) \(\frac{4}{9}\)
D) \(\frac{1}{9}\)
E) \(\frac{7}{9}\)
Three collinear points \(A, B, C\) are such that \(B\) is between \(A\) and \(C\). The distance from \(A\) to \(B\) is 8 more than 4 times the distance from \(B\) to \(C\), and the distance from \(A\) to \(C\) is 5 times the distance from \(B\) to \(C\). What is the distance from \(A\) to \(C\)?
A) 4
B) 20
C) 16
D) 24
E) 12