Practice (90)

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The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property?

For how many positive integers $n$ does $1+2+...+n$ evenly divide from $6n$?

Let $S$ be the set of the $2005$ smallest positive multiples of $4$, and let $T$ be the set of the $2005$ smallest positive multiples of $6$. How many elements are common to $S$ and $T$?

For each positive integer $m > 1$, let $P(m)$ denote the greatest prime factor of $m$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?

How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$?

For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \ldots + n$?

Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^2 - y^2 = m^2$ for some positive integer $m$. What is $x + y + m$?

What is the largest prime that divides both $20! + 14!$ and $20!-14!$?

Using each of the digits 1 to 6, inclusive, exactly once, how many six-digit integers can be formed that are divisible by 6?

What is the units digit of the product $7^{23} \times 8^{105} \times 3^{18}$?

If $12_3$ + $12_5$ + $12_7$ + $12_9$ + $12_x$ = $101110_2$ , what is the value of $x$, the base of the fifth term?

Let $\triangle ABC$ be a right triangle whose three sides' lengths are all integers. Prove among its three sides' lengths, at lease one is a multiple of $3$, one is a multiple of $4$, and one is a multiple of $5$. (Note: they can be the same side. For example, in the $5-12-13$, $12$ is both a multiple of $3$ and $4$.)


How many positive two-digit integers have exactly $8$ positive factors?

When $(37 \times 45) - 15$ is simplified, what is the units digit?

Farmer Hank has fewer than $100$ pigs on his farm. If he groups the pigs five to a pen, there are always three pigs left over. If he groups the pigs seven to a pen, there is always one pig left over. However, if he groups the pigs three to a pen, there are no pigs left over. What is the greatest number of pigs that Farmer Hank could have on his farm?

What fraction of the first 100 triangular numbers are evenly divisible by 7? Express your answer as a common fraction.

The sum of three primes is 125. The difference between the largest and the smallest is 50. What is the largest possible median of these three prime numbers?

We have a set of numbers {1, 2, 3, 4, 5} and we take products of three different numbers. We must find now many pairs of relatively prime numbers there are.

What is the largest five-digit integer such that the product of the digits is $2520$?

For how many two-element subsets {$a,b$} of the set {$1, 2, 3, \cdots, 36$} is the product of $ab$ a perfect square?

What percent of the positive integers $\le$ 36 are factors of 36?

$S$ and $T$ are both two-digit integers less than 80. Each number is divisible by 3. $T$ is also divisible by 7. $S$ is a perfect square. $S + T$ is a multiple of 11, so what is the value of $T$?

How many positive integers not exceeding $2000$ have an odd number of factors?

Find the smallest positive integer $n$ such that 20 divides $15n$ and 15 divides $20n$.

Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube.