AMC8 2027 - Self-practice Set 10

Question 1

What is the sum of all two-digit positive integers with exactly 12 positive divisors?

Question 2

How many three-element subsets of $\{89,95,99,132,166,173\}$ have even sum?

Question 3

For a five-digit integer $n$ letq andr be the quotient and remainder upon division by 100. How many values ofn haveq+r divisible by 11?

8181
9000
4500
1111
3060

Question 4

Use the three vertices and three side midpoints of an equilateral triangle. How many similarity classes of nondegenerate triangles can be formed?

Question illustration

Question 5

Two consecutive positive integers have sum less than 100. Which listed number could be the positive difference of their squares?

Question 6

How many positive integers $n$ maken²−3 n+2 a prime?

0
1
2
Infinitely many
More than 2 but finitely many

Question 7

Find the largest positive integer that leaves the same remainder when dividing 13511,13903,and 14589.

28
an odd multiple of 7 greater than 49
49
98
an even multiple of 7 greater than 98

Question 8

How many trailing zeros does 25^7·8^3 have?

Question 11

What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}?$

$ 4 $
$ 4\sqrt{2} $
$ 8 $
$ 8\sqrt{2} $
$ 16 $

Question 12

How many positive integers can fill the blank in the sentence below?

“One positive integer is ___ more than twice another, and the sum of the two numbers is $28$”

$6$
$7$
$8$
$9$
$10$

Question 13

What is the measure of the acute angle formed by the hands of a clock at $4{:}20$ a.m.?

$0^{\circ}$
$5^{\circ}$
$8^{\circ}$
$10^{\circ}$
$12^{\circ}$

A rectangular garden $50$ feet long and $10$ feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

$100$
$200$
$300$
$400$
$500$

Question 15

Ms. Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?

$ \ 76 $
$ \ 120 $
$ \ 128 $
$ \ 132 $
$ \ 136 $

Question 16

How many whole numbers between $100$ and $400$ contain the digit $2?$

$100$
$120$
$138$
$140$
$148$

Question 17

Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen?

$1$
$3$
$6$
$10$
$12$

Question 18

Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76,$ $94,$ and $87.$ In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests?

$48$
$52$
$66$
$70$
$74$

Question 19

Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6.$ What is the probability that Diana’s number is larger than Apollo’s number?

$\dfrac13$
$\dfrac{5}{12}$
$\dfrac49$
$\dfrac{17}{36}$
$\dfrac12$

Question 20

Which of the following is the correct order of the fractions $\dfrac{15}{11},$ $\dfrac{19}{15},$ and $\dfrac{17}{13},$ from least to greatest?

$\dfrac{15}{11} \lt \dfrac{17}{13} \lt \dfrac{19}{15}$
$\dfrac{15}{11} \lt \dfrac{19}{15} \lt \dfrac{17}{13}$
$\dfrac{17}{13} \lt \dfrac{19}{15} \lt \dfrac{15}{11}$
$\dfrac{19}{15} \lt \dfrac{15}{11} \lt \dfrac{17}{13}$
$\dfrac{19}{15} \lt \dfrac{17}{13} \lt
\dfrac{15}{11}$