- Simplify $$\sqrt{\sqrt{9}-\sqrt{8}}$$
- Simplify $$\sqrt{\sqrt{4}-\sqrt{3}}$$
- Simplify $$\sqrt{\sqrt{25}-\sqrt{24}}$$
- Simplify $$\sqrt{\sqrt{49}-\sqrt{48}}$$
- Simplify $$\sqrt{\sqrt{64}-\sqrt{63}}$$
- Simplify $$\sqrt{\sqrt{225}+\sqrt{224}}$$
- Simplify $$\sqrt{\sqrt{16}+\sqrt{15}}$$
- Simplify $$\sqrt{\sqrt{5}+\sqrt{4}}$$
- Simplify $$\sqrt{\sqrt{81}+\sqrt{80}}$$
- Simplify $$\sqrt{\sqrt{121}+\sqrt{120}}$$
- ________ What is the sum of all positive two-digit integers with exactly 12 positive factors? $$ $$
- ________ What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors? $$ $$
- ________ The sequence 3, 4, 13, 15, 30, 33, 54, 58, 85, 90, …. contains, in increasing order, all the positive integers that yield a triangle number when multiplied by 7. What is the $100^{th}$ term of this sequence? $$ $$
- ________ How many whole numbers $n$, such that $100 \le n \le 1000$, have the same number of odd factors as even factors? $$ $$
- ________ How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12$.$$ $$
- ________ Let $(a\times b\times c)\div(a+b+c)=341$ to be an equation where $a$, $b$ and $c$ are consecutive positive integers. What is the least possible value of $a$? $$ $$
- ________ The letters $A$, $B$, $C$, $D$, $E$ and $F$ represent digits and $ABC,DEF$ represents a positive six-dight integer. What is the number $ABC,DEF$ if $$4(ABC,DEF)=3(DEF,ABC)$$ $$ $$
- ________ Jan is thinking of a positive integer. Her integer has exactly 16 positive divisors, two of which are 12 and 15. What is Jan’s number? $$ $$
- ________ Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady’s $2010^{th}$ step? $$ $$
- ________ The product of a set of positive integers is 144. What is the least possible sum of this set of positive integers? $$ $$
- ________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$
- ________ A digit can be placed in each of the boxes for hundreds and units digits to form the least positive five-digit number divisible by 96. What is the ratio between the smaller digit to the larger digit? Express your answer as a common fraction. $$21\boxed{\phantom{8} }7\boxed{\phantom{8}}$$
- ________ In order, the first four terms of a sequence are 2, 6, 12 and 72, where each team, beginning with the third term, is the product of the two proceeding terms. If the ninth term is $2^a3^b$, what is the value of $a+b$? $$ $$
- ________ What is the sum of the numbers less than 200 that have exactly 9 divisors? $$ $$
- ________ Given that $A=\{\dfrac{n}{24}\}$, $n$ is a natural number, $gcd(n,24)=1$, and $\dfrac{n}{24}\lt 2$, what is the sum of the elements in $A$? $$ $$
- ________ What is the sum of all integer values of $n$ such that $\dfrac{20}{2n-1}$ is an integer?
- ________ What is the units digit of the sum of the sum of all the integers from 100 to 202 inclusive?
- ________ How many of the divisors of $8!$ are larger than $7!$?
- ________ If $n$ is a prime number, what is the smallest composite number produced by $n^2-n-1$?
- ________ $A$, $B$, $C$, and $D$ are distinct positive integers such that the product $AB=60$, the product $CD=60$ and $A-B=C+D$. What is the value of $A$?
