Number Theory
Find the remainder when 1234 is divided by 9.
What is 1?
To find a remainder when a # is divided by 9 we can sum digits of the number and find the remainder of sum divided by 9
A jar contains 3 red marbles, 2 blue marbles, and 1 green marble. If one marble is randomly selected from the jar, what is the probability that it is red?
What is 1/2?
Total number of marbles = 3 red + 2 blue + 1 green = 6. The probability of selecting a red marble is the ratio of red marbles to the total marbles:
P(red)= 3/6 = 1/2
In triangle ABC, the lengths of sides AB and AC are 5 and 6, respectively. If the length of side BC is 7, what is the area of triangle ABC?
What is 6 root 6?
Using Heron's formula, we first find the semi-perimeter s:
s=AB+AC+BC / 2
5+6+7 / 2 = 9
Now, we apply Heron’s formula:
Area= square root of s(s−a)(s−b)(s−c)
The sum of two natural numbers is 17,402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
A) 10,272 B) 11,700 C) 13,362 D) 14,238 E)15,426
What is D?
Let's denote the two natural numbers as x and y, where x is the number that is divisible by 10, and y is the other number. According to the problem, we have the following information:
The sum of the two numbers:
x+y=17,402The number x is divisible by 10, which means it can be expressed as:
x=10k for some integer kIf the units digit of x is erased, the remaining number equals y: y=k
Substituting y=k into the sum equation: 10k+k=17,402 11k=17,402. solve for k
k = 1,581 find x and y
Now, we need to find the difference x−y:
x−y=15,810−1,581Calculating this gives:
x−y=14,229
10k+k=17,40
Menkara has a 4x6 index card. If she shortens the length of one side of this card by 1 inch, the card would have an area 18 square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by 1 inch?
A) 16 B) 17 C)18 D)19 E)20
What is E?
Give the greatest common divisor of 56 and 98?
What is 14?
One way of solving this is by using the Euclidean algorithm:
gcd(98,56) = gcd(98-56, 56) = gcd(42-56)
gcd(56,42) = 14
A jar of M&Ms contains 23 red M&Ms, 45 yellow, 31 green, 12 blue and 21 orange, if you eat 1 of them what is the probability that you ate a blue one?
What is 9.09%
Step 1: Total M&Ms in the jar
The total number of M&Ms in the jar is:
23+45+31+12+21=132
Now, the probability of eating a blue M&M is:
12/132
Simplify the fraction:
12/132=1/11
So, the probability of eating a blue M&M is 1/11, or approximately 0.0909 (9.09%).
A rectangle has a length that is 3 times its width. If the perimeter of the rectangle is 48 units, what are the dimensions of the rectangle?
What is 18 units (length) and 6 units (width)?
The formula for the perimeter P of a rectangle is given by:
P=2l+2w
Substituting the known perimeter of 48 units, we have:
48=2(3w)+2w
48=8w
Which following is a perfect square?
A. (14!*15!)/2 B. (15!*16!)/2 C.(16!*17!)/2 D.(17!*18!)/2 E.(18!*19!)/2
D. (17! * 18!)/2
For (17!18!)/2
(17!)2 * 32
Here, this is a perfect square since it can be expressed as a square.
What is the value of (22 - 2)-(32 - 3)+(42 - 4)?
A)1 B)2 C)5 D)8 E)12
What is D?
calculate each term
State the number of integers between 1 and 1000 that are divisible by 2,3 or 5.
What is 734?
Use the principle of inclusion-exclusion:
#'s divisible by 2: 1000/2 = 500
#'s divisible by 3: 1000/3 = 333
#'s divisible by 5: 1000/5 = 200
Now subtract those divisible by the least common multiples:
#'s divisible by 6(LCM of 2 and 3): 1000/6 = 166
#'s divisible by 10(LCM of 2 and 5): 1000/10 = 100
#'s divisible by 15(LCM of 3 and 5): 1000/15 = 66
#'s divisible by 30(LCM of 2,3 and 5): 1000/30 = 33
500 + 333 + 200 - 166 - 100 -66 + 33 = 734
A bag contains 5 red marbles and 7 blue marbles. Two marbles are drawn at random, one after the other, without replacement. What is the probability that the two marbles drawn are of different colors?
What is 35/66?
In a circle, two chords AB and CD intersect at point P. If AP = 6, PB = 4, and CP = 3, what is the length of PD?
What is 8?
Given Values
Step 1: Set Up the Equation
Using the intersecting chords theorem:
AP⋅PB=CP⋅PD
Substituting the known values:
6⋅4=3⋅x
Calculating the left side:
24=3x
x=24/3=8
The weight of 1/3 of a large pizza together with 3 1/2 cups of orange slices is the same as the weight of 3/4 of a large pizza with 1/2 cup of orange slices. A cup of orange slices is 1/4 of a pound. What is the weight in pounds of a large pizza?
A) 1 4/5 lbs
1/3p+7/8=3/4p+1/8
Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 students. How many students does Portia's high school have?
A) 600 B)650 C) 1950 D)2000 E)2050
What is C?
eq. L+3L=2600
solve for L
then
3L=3×650=1950
Find the largest integer n such that 7n divides 100!
What is 16?
To find the largest integer n such that 7^n divides 100!, we need to determine how many factors of 7 are present in 100!.
This can be done using Legendre’s (or de Polignac's) formula, which counts the number of times a prime p divides n!. For a given prime p, the formula is:
Largest integer n such that pn divides n!=∑k=1∞⌊npk⌋Largest integer n such that pn divides n!=k=1∑∞⌊pkn⌋
Here, p=7 and n=100.
[100/7] = 14
[100/72] = [100/49] = 2
[100/73] = [100/343] = 0
Since 73 = 343 is larger than 100, we stop here.
14 + 2 = 16
In a class of 30 students, each student has a different favorite fruit among apples, oranges, and bananas. The distribution of their favorite fruits is as follows: 12 students favor apples. 10 students favor oranges. The remaining students favor bananas. If a student is chosen at random, what is the probability that this student either favors apples or bananas?
What is 2/3?
In triangle ABC, angle A measures 60 degrees, and the length of sides AB and AC are 8 and 10. What is the length of side BC?
What is 2√21?
use the cosine rule c2=a2+b2−2abcosC
A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For example, Arbor Day last year was displayed as 20230428. For how many dates in 2023 does each digit appear an even number of times in the 8-digit display for that date?
What is the maximum number of balls of clay with radius 2 that can completely fit inside a cube of side length 6, assuming that the balls can be reshaped but not compressed before they are packed in the cube?
A) 3 B) 4 C) 5 D) 6 E) 7
What is D?
he volume Vcube of a cube is given by the formula:
Vcube=side length^3=6^3=216 cubic units
Vball=4/3πr^3
substitute 2 in for r
Maximum number of balls= 81/4*pi
plug in 3.14 for pi
81/12.56 = 6.45 (int) = 6
Find the sum of all positive integers less than 1000 that are relatively prime to 1000.
What is 200000?
Using Euler's Totient Function and Sum Formula for Numbers Relatively Prime to n
In a game, you flip a fair coin until you get two consecutive heads (HH). What is the expected number of flips required to achieve this outcome?
What is 6?
In triangle ABC, angle A measures 45 degrees, and the circumradius of the triangle is 10. If BC = 12, what is the area of the triangle ABC?
What is 72?
Circles C1 and C2 each have a radius 1, and their distance between their centers is 1/2. Circle 3 is the largest circle internally tangent to both C1 and C2. Circle C4 is internally tangent to C1 and C2 and externally tangent to C3. What is the radius of C4?
A. 1/14 B. 1/12 C. 1/10 D. 3/28 E. 3/18
The area of the region bounded by the graph of
x2 + y2 = 3 l x-y l + 3 l x+y l
is m + n*pi, where m and n are integers. What is m + n? (consider the entire covered effective area)
A) 18 B)27 C)36 D)45 E)54
What is E?