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Lily
Questions (894)
If catalhuyuk had been built near a major system how might life have been different
1 answer
11 views
Which term describes the repeated arrangement of the same molecule?
A. molecular model B. Atoms C. Bonds D. extended structure
1 answer
25 views
In base $b,$ the square of the number $17_b$ is $211_b$. Find the base $b$.
1 answer
92 views
The equation y = -16t^2 + 26t + 105 describes the height (in feet) of a ball tossed up in the air at 26 feet per second from a
1 answer
37 views
Let $m$ and $n$ be non-negative integers. If $m = 6n + 2$, then what integer between $0$ and $m$ is the inverse of $5$ modulo
1 answer
42 views
Find the remainder when 40^{13} is divided by 81.
1 answer
34 views
Let $p$ be a prime. What are the possible remainders when $p$ is divided by $17?$ Select all that apply.
1 answer
26 views
For every integer n≥1, let P(n) denote the product of the first n odd numbers:
P(n)=1×3×5×⋯×(2n−1) Prove that for all
1 answer
30 views
For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when divided by $23.$ What remainder does
1 answer
45 views
For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) +
1 answer
28 views
The Fibonacci sequence is defined by F_1 = F_2 = 1 and F_{n + 2} = F_{n + 1} + F_n. Find the remainder when F_{1000} is divided
1 answer
50 views
The sequence $a_1$, $a_2$, $a_3,$ $\dots$ is defined by $a_1 = 1,$ $a_2 = 3,$ and
a_n = 2a_{n - 1} + a_{n - 2}. Thus, the next
1 answer
52 views
The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2. Compute the ordered triple of
1 answer
38 views
Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.
1 answer
25 views
Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$
1 answer
52 views
In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at
1 answer
34 views
Let $x$ and $y$ be real numbers such that $x^3 + 3xy^2 = 679$ and $x^3 - 3xy^2 = 615.$ Find $x - y.$
1 answer
62 views
Fill in the blanks to get an identity. (Each blank should contain an integer.)
(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k +
1 answer
44 views
(a) Let $ABC$ be an equilateral triangle, centered at $O.$ A point $P$ is chosen at random inside the triangle. Find the
1 answer
30 views
(a) Find the last two digits of $9*7*8$.
(b) Find the last two digits of $2^{3^4}$. (By convention, exponent towers are evaluated
1 answer
36 views
Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by $2$.
1 answer
35 views
Calculate $40*9 \pmod{23}.$ Express your answer as a non-negative integer that is less than $23$.
1 answer
29 views
Find the remainder when $100^{100}$ is divided by $18$.
1 answer
41 views
Let $a$ be an integer such that $a \equiv 5 \pmod{7}$. Find the value of $a + 1 \pmod{7}$. Express your answer as a residue
1 answer
42 views
Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.
1 answer
29 views
A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are $7$,
1 answer
52 views
Which of the residues $0,$ $1,$ $2,$ $3,$ $4,$ $5$ satisfy the congruence $x^2 \equiv 0 \pmod{6}?$
Give your answer as a list,
1 answer
27 views
Which of the residues $0,$ $1,$ $2,$ $3,$ $4$ satisfy the congruence $3x \equiv 0 \pmod{5}?$
Give your answer as a list,
1 answer
27 views
Which of the residues $0,$ $1,$ $2,$ $\dots,$ $11$ satisfy the congruence $3x \equiv 8 \pmod{5}?$
Give your answer as a list,
1 answer
27 views
The following cards are split into three piles at random, so that every pile contains the same number of cards. What is the
1 answer
81 views
Alex chooses a number at random from the set $\{1, 2, 3, \dots, 5\}.$ Winnie also chooses a number at random from the same set.
1 answer
32 views
Two different numbers are chosen at random from the set {1, 2, 3, ..., 5}. What is the probability that the product of the
3 answers
61 views
On planet Enigma, the residents use a currency called the confusion. There are only two confusion bills on Enigma, one worth $7$
1 answer
33 views
Let $a,$ $b,$ $c,$ and $n$ be positive integers. If a + b + c = 793 and
\[a + 5n = b - 3n = cn,\] compute the value of $a.$
1 answer
39 views
Find the largest positive integer $n$ such that
\[\frac{(n + 1)^2}{n + 2}\] is an integer.
1 answer
30 views
The product of a set of distinct positive integers is $630$. If one of the numbers is 42, what are the other two numbers?
1 answer
54 views
How many positive four-digit integers $N$ have the property that the three-digit number obtained by removing the leftmost digit
1 answer
67 views
What is the largest integer $n$ such that $11^n$ divides $1,000,000!$ ?
1 answer
45 views
Starting with the A and moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell
1 answer
38 views
A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways
1 answer
57 views
A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways
1 answer
46 views
A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways
1 answer
39 views
Starting with the A moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
1 answer
67 views
Find the number of positive integers that satisfy both the following conditions:
Each digit is a 1 or a 3. The sum of the digits
1 answer
45 views
The sum of the reciprocals of three positive integers is $\frac{13}{12}$. What are the three positive integers?
1 answer
32 views
Starting at the same time, Bill rings a bell every $36$ seconds, and Wendy blows a whistle every $90$ seconds, and Sam blows a
1 answer
30 views
Starting at the same time, Bill rings a bell every $36$ seconds, and Wendy blows a whistle every $1000$ seconds. How many
1 answer
29 views
How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $21?$
1 answer
48 views
Annville Junior High School has a math club and a science club. There are $3$ students in the math club and $3$ students who are
1 answer
48 views
Annville Junior High School has a math club and a science club. There are $3$ students in the math club and $3$ students who are
1 answer
42 views
When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is
1 answer
33 views
Solve for the variable x in terms of y and z:
xy + 2x = x - 2y + z + xz - 5xy + x^2 - x^3*y + x*y^2.
1 answer
26 views
Compute gcd of 702,936 and 841,131.
1 answer
31 views
Compute gcd of 294,343 and 878,568.
1 answer
30 views
What is the smallest prime divisor of 5^{19} + 7^{13} + 23?
1 answer
27 views
Compute gcd of 123,451 and 708,677.
1 answer
27 views
Compute gcd of 36,104 and 22,508.
1 answer
27 views
In how many ways can $4$ balls be placed in $3$ boxes if the balls are distinguishable, and the boxes are indistinguishable, and
3 answers
55 views
In how many ways can $4$ balls be placed in $3$ boxes if the balls are indistinguishable, and the boxes are distinguishable, and
1 answer
49 views
In how many ways can $4$ balls be placed in $3$ boxes if neither the balls nor the boxes are distinguishable, and no box can be
1 answer
35 views
In how many ways can $4$ balls be placed in $3$ boxes if the balls and boxes are both distinguishable, and no box can be empty?
1 answer
42 views
In how many ways can $4$ balls be placed in $3$ boxes if the balls and boxes are both distinguishable, and no box can be empty?
I
1 answer
39 views
When fractions are expressed in different bases, they can be terminating or repeating. For example, when \frac{1}{5} is
1 answer
32 views
Find the greatest prime divisor of the value of the arithmetic series
1 + 2 + 3 + \dots + 135 + 136 + 137 + 138 + 139 + 140.
1 answer
45 views
Find the 4000th digit following the decimal point in the expansion of \frac{1}{17}.
1 answer
31 views
What is the smallest prime divisor of 5^{19} * 7^{13} * 3^{31}?
1 answer
31 views
How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?
1 answer
32 views
What is the fifth smallest prime number greater than 2000?
1 answer
32 views
Find the sum of the distinct prime factors of 5005 + 6192 + 11823.
1 answer
34 views
Find the sum of the distinct prime factors of 781 + 691 + 3858.
1 answer
33 views
Find the sum of the distinct prime factors of 728 + 118 + 2024.
1 answer
30 views
Find the sum of the distinct prime factors of 248 + 113 + 2870.
1 answer
35 views
Find the sum of the distinct prime factors of 333 + 183 + 401.
1 answer
30 views
Find the sum of the distinct prime factors of 123 + 456 + 789.
1 answer
32 views
Some of the following integers are prime. Select all the prime integers in the list.
1 2 3 4 5 6 7 8 9 10
1 answer
28 views
What is the fifth smallest prime number greater than 500?
1 answer
31 views
What is the fifth smallest prime number greater than 100?
1 answer
28 views
Jon teaches a fourth grade class at an elementary school where class sizes are always at least 10 students and at most 15. One
1 answer
47 views
What are the first 5 digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{7}$ is
1 answer
27 views
Express 2.\overline{5}_{71} as a base 10 fraction in reduced form.
1 answer
33 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in
1 answer
40 views
What is 100000_{16} * E21A8_{16} in base 16?
1 answer
26 views
Express 0.1101_{25} as a base 10 fraction in reduced form.
1 answer
35 views
Express 0.\overline{212}_{30} as a base 10 fraction in reduced form.
1 answer
35 views
Compute A710_{16} + 51_{16} + B8AE_{16} in base 16.
1 answer
27 views
Compute 221_3 + 122_3 + 202_3 + 110_3 in base 3.
1 answer
29 views
Compute 25_6 + 54_6 + 512_6 + 401_6 in base 6.
1 answer
28 views
Compute 35A8_{11} * 2751_{11} * A04_{11}. Give your answer as a base 11 integer.
1 answer
45 views
Compute 11_2 + 110_2 + 1111_2 + 11111_2 + 1101_2 + 10100_2. Give your answer as a base 2 integer.
3 answers
51 views
Compute 2A31_{12} * B75_{12} * 88_{12}. Give your answer as a base 12 integer.
3 answers
42 views
Compute 357_8 * 223_8 * 11_8. Give your answer as a base 8 integer.
1 answer
24 views
Find the smallest positive integer B so that when we express the decimal number 99 as a base B number, we still get a 2-digit
1 answer
39 views
What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{3}$ is
1 answer
24 views
Express 2.\overline{57} as a base 10 fraction in reduced form.
1 answer
44 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in
1 answer
40 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $11.$ When the number $N$ is expressed in
1 answer
56 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $11.$ When the number $N$ is expressed in
1 answer
53 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $11.$ When the number $N$ is expressed in
1 answer
44 views
Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $11.$ When the number $N$ is expressed in
1 answer
45 views
Express 0.11012 as a base 10 fraction in reduced form.
1 answer
31 views
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