Practice (91)
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Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
How many three-digit numbers are divisible by 13?
How many two-digit numbers have digits whose sum is a perfect square?
What is the sum of the two smallest prime factors of $250$?
Tiles $I, II, III$ and $IV$ are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
Pick two consecutive positive integers whose sum is less than $100$. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil?
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
Prove: for any given positive integer $n$, the value of $(n+7)^2 -(n-5)^2$ must be a multiple of 24.
What is the greatest prime factor of the product $6 \times 14 \times 22 $?
Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.
Prove: randomly select $51$ numbers from $1$, $2$, $3$, $\dots$, $100$, there must exist two numbers for which one is a multiple of the other.
Prove: it is impossible to have two positive integers such that the product of their sum and their difference equals 1990.
17 people attend a party. Prove: it is impossible that everyone exactly shakes hands with 3 other attendees.
Let $n>1$ be a positive integer. Prove $1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}$ cannot be a whole integer.
Find all orders combination of digits $A$, $B$, and $C$ such that the 6-digit number $\overline{503ABC}$ is a multiple of 7, 9, and 11.
What is the sum of the prime factors of $2010$?
How many digits are in the product $4^5 \cdot 5^{10}$?
In how many ways can $10001$ be written as the sum of two primes?
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$?