Tüm alıştırma soruları

Soru 201Soru →

Let $n$ be a positive integer with exactly two distinct prime factors. If $n$ is a multiple of $12$ , is not a multiple of $8$ , and has exactly $12$ positive factors, what is the number of positive factors of $n^2$ ?

24

33

35

45

144

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The number of positive factors of

n^2

is 35.

Since

n

is a multiple of

12 = 2^2 \cdot 3^1

and has exactly two distinct prime factors, its prime factorization must be

n = 2^a \cdot 3^b

. The condition that

n

is not a multiple of

8 = 2^3

forces

a = 2

. Using the factor count formula

(a+1)(b+1) = 12

, we get

(2+1)(b+1) = 12

, which gives

b = 3

. Therefore,

n^2 = (2^2 \cdot 3^3)^2 = 2^4 \cdot 3^6

. The total number of positive factors of

n^2

is

(4+1)(6+1) = 35

.

Adım Adım Çözüm

1

Determine the prime factors of

n

n = 2^a \cdot 3^b

, where

a \geq 2

and

b \geq 1

Since

n

has exactly two distinct prime factors and

12 = 2^2 \cdot 3^1

divides

n

, the only prime factors of

n

are

2

and

3

.

2

Apply the divisibility constraints on the exponent

a

a = 2

Since

12

divides

n

,

a \geq 2

. Since

n

is not a multiple of

8 = 2^3

,

a < 3

. Thus

a

must equal

2

.

3

Use the factor count formula to find

b

b = 3

, so

n = 2^2 \cdot 3^3

The number of positive factors of

n

is

(a+1)(b+1) = (2+1)(b+1) = 12 \implies 3(b+1) = 12 \implies b = 3

.

4

Calculate the number of positive factors of

n^2

(4+1)(6+1) = 35

Squaring

n

gives

n^2 = (2^2 \cdot 3^3)^2 = 2^4 \cdot 3^6

. By the factor count formula,

n^2

has

(4+1)(6+1) = 35

positive factors.

Anahtar Kavram

Factor Count Formula & Prime Factorization Constraints

Tahmini Süre:1m 45s

Soru 202Soru →

If $N = 6^6 + 6^7 + 6^8 + 6^9$ , what is the sum of the exponents of all prime factors in the prime factorization of $N$ ?

14

30

15

8

13

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

Factoring out

6^6

gives

N = 6^6(1 + 6 + 36 + 216) = 6^6(259)

. Decomposing composite bases into prime factors yields

6^6 = 2^6 \times 3^6

and

259 = 7^1 \times 37^1

. Thus, the prime factorization is

N = 2^6 \times 3^6 \times 7^1 \times 37^1

. Summing the prime exponents gives

6 + 6 + 1 + 1 = 14

.

Adım Adım Çözüm

1

Factor out the common term

6^6

from the expression.

N = 6^6(1 + 6 + 6^2 + 6^3) = 6^6(1 + 6 + 36 + 216) = 6^6 \times 259

Factoring simplifies the sum into a single product of terms.

2

Express

6^6

in terms of its prime factors.

6^6 = (2 \times 3)^6 = 2^6 \times 3^6

The base 6 is composite and must be broken down into prime factors 2 and 3.

3

Determine the prime factorization of 259.

Testing small primes shows

259 = 7 \times 37

, where both 7 and 37 are prime numbers.

259 is not prime and must be decomposed into its prime components

7^1 \times 37^1

.

4

Combine all prime factors to write the complete prime factorization of

N

.

N = 2^6 \times 3^6 \times 7^1 \times 37^1

Writing

N

in standard canonical form reveals all prime exponents.

5

Sum the exponents of all prime factors.

6 + 6 + 1 + 1 = 14

The question asks for the total sum of the exponents of the prime factors.

Anahtar Kavram

Prime Factorization of Factored Exponential Sums

Tahmini Süre:2m 30s

Soru 203Soru →

At the start of a month, an online bookstore's inventory consisted of Fiction, Non-Fiction, and Textbook titles. Exactly $\frac{3}{8}$ of the total inventory consisted of Fiction titles, and $0.40$ of the remaining inventory consisted of Non-Fiction titles, with the balance consisting of Textbook titles. During the month, the number of Fiction titles increased by $20\%$ , the number of Non-Fiction titles decreased by $25\%$ , and the number of Textbook titles remained unchanged. If the total inventory increased by a net amount of $15$ titles at the end of the month, how many total book titles were in the inventory at the start of the month?

400

600

800

1,200

2,400

Cevabı ve açıklamayı göster

Cevap: 1,200

Cevap

1,200 total book titles

The initial inventory consists of

\frac{3}{8}T

Fiction titles and

0.40 \times \frac{5}{8}T = \frac{1}{4}T

Non-Fiction titles. A

20\%

increase in Fiction adds

\frac{3}{40}T

, while a

25\%

decrease in Non-Fiction subtracts

\frac{1}{16}T

. Combining these gives a net change of

\frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = \frac{1}{80}T

. Setting

\frac{1}{80}T = 15

gives

T = 1,200

.

Adım Adım Çözüm

1

Determine the initial fractional breakdown of each book category relative to the total inventory T

Fiction

= \frac{3}{8}T

. Remaining inventory

= 1 - \frac{3}{8} = \frac{5}{8}T

. Non-Fiction

= 0.40 \times \frac{5}{8}T = \frac{2}{5} \times \frac{5}{8}T = \frac{1}{4}T

.

The question specifies that Non-Fiction is 0.40 of the remaining inventory, not the total inventory.

2

Calculate the net change in titles as a fraction of total initial inventory T

Increase in Fiction

= 20\% \times \frac{3}{8}T = \frac{1}{5} \times \frac{3}{8}T = +\frac{3}{40}T

.
Decrease in Non-Fiction

= 25\% \times \frac{1}{4}T = \frac{1}{4} \times \frac{1}{4}T = -\frac{1}{16}T

.
Net fractional change

= \frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = +\frac{1}{80}T

.

Percentage changes must be applied to each category's specific share of the total inventory.

3

Equate the net fractional change to the numerical net increase and solve for T

\frac{1}{80}T = 15 \implies T = 15 \times 80 = 1,200.

The overall net increase is given as 15 titles.

Anahtar Kavram

Multi-step successive fraction and percentage change with changing base values

Tahmini Süre:2m 0s

Soru 204Soru →

What is the value of $(2^3 + 2^3)^2$ ?

2^4

2^7

2^8

2^{12}

2^{18}

Cevabı ve açıklamayı göster

Cevap: $2^8$

Cevap

The correct answer is

2^8

.

Combining the terms inside the parentheses gives

2^3 + 2^3 = 2 \cdot 2^3 = 2^4

. Squaring

2^4

using the power rule

(a^m)^n = a^{m \cdot n}

produces

2^{4 \cdot 2} = 2^8

.

Adım Adım Çözüm

1

Simplify the expression inside the parentheses

2^3 + 2^3 = 2 \cdot (2^3) = 2^1 \cdot 2^3 = 2^4

Adding two identical terms is equivalent to multiplying the term by 2, which allows combining powers of the same base.

2

Apply the power of a power exponent rule to the outer square

(2^4)^2 = 2^{4 \times 2} = 2^8

When raising a power to another power, multiply the exponents:

(a^m)^n = a^{m \cdot n}

.

Anahtar Kavram

Combining like exponential terms and applying power rules

Tahmini Süre:45s

Soru 205Soru →

For positive integers $a$ , $b$ , and $c$ , let $x = 2^a \times 3^3 \times 5^b$ and $y = 2^3 \times 3^c \times 5^1$ . If $\text{GCD}(x, y) = 360$ and $\text{LCM}(x, y) = 270,000$ , what is the value of $a + b + c$ ?

8

9

10

11

12

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

First, express the given GCD and LCM in prime factor form:

360 = 2^3 \times 3^2 \times 5^1

and

270,000 = 2^4 \times 3^3 \times 5^4

. For any two numbers, the exponent of each prime factor in their GCD is the minimum of their individual exponents, while the exponent in their LCM is the maximum. Comparing prime 2:

\max(a, 3) = 4

, so

a = 4

. Comparing prime 3:

\min(3, c) = 2

, so

c = 2

. Comparing prime 5:

\max(b, 1) = 4

, so

b = 4

. Therefore,

a + b + c = 4 + 4 + 2 = 10

.

Adım Adım Çözüm

1

Find the prime factorizations of GCD(x, y) and LCM(x, y)

360 = 2^3 × 3^2 × 5^1 and 270,000 = 2^4 × 3^3 × 5^4

Expressing GCD and LCM in prime factor form allows direct comparison of prime exponents.

2

Apply prime exponent rules for GCD (minimum) and LCM (maximum)

For prime 2: min(a, 3) = 3 and max(a, 3) = 4, so a = 4.
For prime 3: min(3, c) = 2 and max(3, c) = 3, so c = 2.
For prime 5: min(b, 1) = 1 and max(b, 1) = 4, so b = 4.

The GCD takes the minimum exponent for each prime factor, while the LCM takes the maximum exponent.

3

Calculate the sum a + b + c

4 + 4 + 2 = 10

Substitute the evaluated exponent values to find the requested total.

Anahtar Kavram

GCD and LCM via Prime Factorization Exponents

Tahmini Süre:1m 30s

Soru 206Soru →

A positive integer $n$ has a prime factorization of the form $2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. If $n$ is a multiple of $20$ but not a multiple of $40$ , is not divisible by $9$ , and has exactly $12$ positive integer divisors, what is the sum of all possible values of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 560

Cevap

The sum of all possible values of

n

is 560.

To determine the sum of all possible values of

n

, first analyze the given conditions on the prime exponents of

n = 2^a \cdot 3^b \cdot 5^c

. Divisibility by

20 = 2^2 \cdot 5^1

requires

a \ge 2

and

c \ge 1

. Since

n

is not divisible by

40 = 2^3 \cdot 5^1

, the exponent

a

must equal 2. Since

n

is not divisible by

9 = 3^2

, the exponent

b

must be strictly less than 2, meaning

b

can be 0 or 1. The total number of positive integer divisors is given by

(a+1)(b+1)(c+1) = 12

. Substituting

a = 2

gives

3(b+1)(c+1) = 12

, or

(b+1)(c+1) = 4

. If

b = 0

, then

c+1 = 4 \implies c = 3

, which gives

n = 2^2 \cdot 3^0 \cdot 5^3 = 500

. If

b = 1

, then

c+1 = 2 \implies c = 1

, which gives

n = 2^2 \cdot 3^1 \cdot 5^1 = 60

. The sum of these two valid values is

500 + 60 = 560

.

Adım Adım Çözüm

1

Analyze the divisibility constraints to find values/ranges for exponents

a

,

b

, and

c

.

a = 2

,

c \ge 1

, and

b \in \{0, 1\}

.

Divisibility by 20 (

2^2 \cdot 5^1

) requires

a \ge 2

and

c \ge 1

. Non-divisibility by 40 (

2^3 \cdot 5^1

) restricts

a < 3

, so

a = 2

. Non-divisibility by 9 (

3^2

) restricts

b < 2

.

2

Apply the divisor count formula

(a+1)(b+1)(c+1) = 12

.

(b+1)(c+1) = 4

.

Plugging

a = 2

into

(a+1)(b+1)(c+1) = 12

yields

3(b+1)(c+1) = 12

, which simplifies to

(b+1)(c+1) = 4

.

3

Evaluate the cases for

b = 0

and

b = 1

.

The valid values for

n

are

500

and

60

.

When

b = 0

,

c+1 = 4 \implies c = 3

, yielding

n = 2^2 \cdot 3^0 \cdot 5^3 = 500

. When

b = 1

,

c+1 = 2 \implies c = 1

, yielding

n = 2^2 \cdot 3^1 \cdot 5^1 = 60

.

4

Calculate the sum of all valid integers

n

.

560

500 + 60 = 560

.

Anahtar Kavram

Divisor Count Formula and Divisibility Constraints

Soru 207Soru →

An interior designer is arranging a row of 7 decorative wall tiles consisting of 3 identical blue tiles, 2 identical yellow tiles, and 2 identical red tiles. If the 2 red tiles cannot be placed next to each other, how many distinct arrangements of the 7 tiles are possible?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The total number of distinct arrangements possible is 150.

To find the total number of distinct arrangements where no two red tiles are adjacent, first calculate the arrangements of the 5 non-restricted tiles (3 blue, 2 yellow), which is

\frac{5!}{3!2!} = 10

. Placing 5 tiles creates 6 available spaces (including the two ends). Selecting 2 of these 6 spaces for the 2 identical red tiles yields

\binom{6}{2} = 15

choices. By the Fundamental Counting Principle, the total number of valid arrangements is

10 \times 15 = 150

.

Adım Adım Çözüm

1

Calculate the number of distinct ways to arrange the non-restricted tiles (3 identical blue and 2 identical yellow).

The number of distinct arrangements of the 5 non-red tiles is

\frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10

.

Arranging all non-restricted tiles first creates the specific positions into which the restricted tiles can be inserted.

2

Determine the number of valid positions for the 2 identical red tiles such that no two are adjacent.

Placing 5 tiles creates 6 available insertion slots (one at each end and four between tiles). Choosing 2 distinct slots out of 6 gives

\binom{6}{2} = \frac{6 \times 5}{2} = 15

ways.

Selecting 2 distinct slots ensures that every selected space holds at most one red tile, guaranteeing that no two red tiles are adjacent.

3

Multiply the results from Step 1 and Step 2 using the Fundamental Counting Principle.

10 \times 15 = 150

.

Each arrangement of non-red tiles can be independently paired with any valid placement of the red tiles.

Anahtar Kavram

Counting arrangements of identical items with non-adjacency restrictions using the slotting method.

Tahmini Süre:1m 30s

Soru 208Soru →

If $x$ is a negative integer and $y$ is a positive integer, what is the value of $\frac{|x - y| - |y - x|}{x + y + 100}$ ?

Cevabı ve açıklamayı göster

Cevap: 0

Cevap

0

Given

x < 0

and

y > 0

, the quantity

x - y

is strictly negative, which means

|x - y| = -(x - y) = y - x

. The quantity

y - x

is strictly positive, so

|y - x| = y - x

. Thus, the numerator simplifies to

(y - x) - (y - x) = 0

. Dividing zero by any non-zero denominator yields

0

.

Adım Adım Çözüm

1

Determine the signs of the terms inside the absolute value functions

Since

x < 0

and

y > 0

,

x - y < 0

and

y - x > 0

.

Subtracting a positive number from a negative number yields a negative value, while subtracting a negative number from a positive number yields a positive value.

2

Simplify the absolute value expressions

|x - y| = y - x

and

|y - x| = y - x

.

The absolute value of a negative number is its negation, and the absolute value of a positive number is the number itself.

3

Compute the difference in the numerator

|x - y| - |y - x| = (y - x) - (y - x) = 0

.

Subtracting an algebraic expression from an identical expression results in zero.

4

Evaluate the entire fraction

\frac{0}{x + y + 100} = 0

.

Zero divided by any non-zero real number is zero.

Anahtar Kavram

Positive and Negative Number Properties with Absolute Value

Soru 209Soru →

Technician A can assemble a solar panel frame in $6$ hours working alone at a constant rate. Technician B can assemble the exact same solar panel frame in $3$ hours working alone at a constant rate. Working together at their respective constant rates, how many hours will it take Technician A and Technician B to assemble one solar panel frame?

1.5

hours

2

hours

3

hours

4.5

hours

9

hours

Cevabı ve açıklamayı göster

Cevap: $2$ hours

Cevap

Working together, Technician A and Technician B will complete the assembly in

2

hours.

The correct option is

2

hours because Technician A completes

\frac{1}{6}

of the work in one hour and Technician B completes

\frac{1}{3}

(or

\frac{2}{6}

) of the work in one hour. Combined, they complete

\frac{3}{6} = \frac{1}{2}

of the work each hour, so the entire job takes

2

hours.

Adım Adım Çözüm

1

Determine individual work rates per hour.

Technician A's rate is

\frac{1}{6}

frame per hour, and Technician B's rate is

\frac{1}{3}

frame per hour.

Work rate is defined as the fraction of a job completed per unit of time, calculated as

\frac{1}{\text{time}}

.

2

Add the individual rates to find the combined rate.

Combined rate

= \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}

frame per hour.

When entities work simultaneously, their individual rates add up.

3

Calculate total time required for one full frame.

Total time

= \frac{1}{\text{Combined Rate}} = \frac{1}{1/2} = 2

hours.

Total time required is the reciprocal of the combined work rate.

Anahtar Kavram

Combined Work Rate Formula

Tahmini Süre:50s

Soru 210Soru →

At the beginning of a fiscal year, a municipal transit authority allocated its capital expenditure budget among three projects: Bus Rapid Transit, Rail Modernization, and Station Upgrades. Exactly $0.30$ of the total budget was allocated to Bus Rapid Transit. Of the remaining budget, exactly $\frac{3}{7}$ was allocated to Rail Modernization, and the rest was allocated to Station Upgrades. By the end of the year, expenditures on Bus Rapid Transit exceeded its initial allocation by $25\%$ , expenditures on Rail Modernization were $15\%$ below its initial allocation, and expenditures on Station Upgrades exceeded its initial allocation by $10\%$ . By what percent did the transit authority's total expenditures across all three projects exceed its initial total budget?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The total expenditures across all three projects exceeded the initial total budget by

7\%

.

To solve this problem, represent the total initial budget as

B

. The Bus Rapid Transit allocation is

0.30B

, leaving

0.70B

. Rail Modernization receives

\frac{3}{7}

of

0.70B

, which equals

0.30B

. The remaining portion for Station Upgrades is

0.70B - 0.30B = 0.40B

. End-of-year expenditures are calculated by multiplying each allocation by its respective growth multiplier: Bus Rapid Transit is

0.30B \times 1.25 = 0.375B

, Rail Modernization is

0.30B \times 0.85 = 0.255B

, and Station Upgrades is

0.40B \times 1.10 = 0.44B

. Summing these expenditures gives

0.375B + 0.255B + 0.44B = 1.07B

. Comparing

1.07B

to the initial

1.00B

reveals an overall increase of

0.07B

, or

7\%

.

Adım Adım Çözüm

1

Express the initial allocations for each project as fractions of the total budget B

Bus Rapid Transit =

0.30B

, Rail Modernization =

0.30B

, Station Upgrades =

0.40B

Bus Rapid Transit is explicitly

0.30B

. The remaining

0.70B

is split such that Rail Modernization receives

\frac{3}{7} \times 0.70B = 0.30B

, leaving

0.70B - 0.30B = 0.40B

for Station Upgrades.

2

Apply the individual percentage changes to determine end-of-year expenditures

Bus Rapid Transit =

0.375B

, Rail Modernization =

0.255B

, Station Upgrades =

0.44B

A

25\%

increase corresponds to a multiplier of

1.25

, a

15\%

decrease corresponds to a multiplier of

0.85

, and a

10\%

increase corresponds to a multiplier of

1.10

.

3

Sum the project expenditures and calculate the net percent change relative to B

Total expenditure =

1.07B

, corresponding to a

7\%

net increase

Adding

0.375B + 0.255B + 0.44B

yields

1.07B

. Subtracting the original budget

1.00B

gives

0.07B

, which is

7\%

of

B

.

Anahtar Kavram

Weighted Percentage Changes and Sequential Fraction-Decimal Operations

Soru 211Soru →

If $x$ is a negative real number and $y$ is a positive real number, what is the value of $\frac{x}{|x|} + \frac{|y|}{y}$ ?

Cevabı ve açıklamayı göster

Cevap: 0

Cevap

The value of the expression is 0.

For any negative number

x

, the ratio

\frac{x}{|x|}

evaluates to

-1

because

|x| = -x

. For any positive number

y

, the ratio

\frac{|y|}{y}

evaluates to

1

because

|y| = y

. Summing

-1

and

1

results in

0

.

Adım Adım Çözüm

1

Evaluate the first term for a negative variable

-1

By definition of absolute value, if

x < 0

, then

|x| = -x

, making

\frac{x}{|x|} = \frac{x}{-x} = -1

.

2

Evaluate the second term for a positive variable

1

If

y > 0

, then

|y| = y

, making

\frac{|y|}{y} = \frac{y}{y} = 1

.

3

Sum the simplified values

0

Combining

-1

and

1

yields

-1 + 1 = 0

.

Anahtar Kavram

Properties of Absolute Value and Signs of Numbers

Soru 212Soru →

If $k$ is a positive integer such that $\sqrt{8^{10} + 2^{23} + 4^7} = 2^k + 128$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

k

is 15.

Converting all terms under the square root to base 2 produces

\sqrt{2^{30} + 2^{23} + 2^{14}}

. Recognizing that

(2^{15} + 2^7)^2 = (2^{15})^2 + 2(2^{15})(2^7) + (2^7)^2 = 2^{30} + 2^{23} + 2^{14}

, taking the square root yields

2^{15} + 2^7 = 2^{15} + 128

. Matching this with

2^k + 128

yields

k = 15

.

Adım Adım Çözüm

1

Convert terms under the square root to base 2.

The radical expression becomes

\sqrt{2^{30} + 2^{23} + 2^{14}}

.

Expressing terms with the same base allows exponent rules and algebraic identity recognition.

2

Identify the expression under the radical as a perfect square of the form

(a + b)^2

.

Setting

a = 2^{15}

and

b = 2^7

gives

2ab = 2 \cdot 2^{15} \cdot 2^7 = 2^{23}

, so

2^{30} + 2^{23} + 2^{14} = (2^{15} + 2^7)^2

.

The middle term

2^{23}

satisfies

2 \cdot \sqrt{2^{30}} \cdot \sqrt{2^{14}}

.

3

Evaluate the square root and solve for

k

.

\sqrt{(2^{15} + 2^7)^2} = 2^{15} + 128

. Setting

2^{15} + 128 = 2^k + 128

yields

k = 15

.

Comparing terms directly after evaluating

2^7 = 128

isolates

2^k = 2^{15}

.

Anahtar Kavram

Application of exponent rules combined with perfect square algebraic identities under radicals

Soru 213Soru →

Pump A can fill a water storage tank in $4$ hours when operating alone at a constant rate. Pump B can fill the same tank in $6$ hours when operating alone at a constant rate. If both pumps operate simultaneously at their respective constant rates, how many hours will it take to fill the empty tank completely?

2.4

hours

5.0

hours

10.0

hours

2.0

hours

\frac{5}{12}

hours

Cevabı ve açıklamayı göster

Cevap: $2.4$ hours

Cevap

2.4

hours

To find the combined time, sum the hourly rates of both pumps:

\frac{1}{4} + \frac{1}{6} = \frac{5}{12}

tanks per hour. Inverting this combined rate gives the total hours required:

\frac{12}{5} = 2.4

hours.

Adım Adım Çözüm

1

Determine individual work rates per hour.

Pump A's rate is

\frac{1}{4}

of the tank per hour, and Pump B's rate is

\frac{1}{6}

of the tank per hour.

Work rate is defined as

\text{Rate} = \frac{\text{Work}}{\text{Time}}

.

2

Add the individual rates to find the combined rate.

\text{Combined Rate} = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}\text{ tank per hour}.

When entities work together, their work rates add linearly.

3

Calculate the total time required for the combined rate to complete

1

full tank.

\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4\text{ hours}.

Time equals total work divided by the combined work rate.

Anahtar Kavram

Combined Work Rate Formula

Tahmini Süre:1m 0s

Soru 214Soru →

How many distinct 4-letter arrangements can be formed by rearranging all of the letters in the word $\text{SEES}$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

To calculate the number of unique arrangements for a multiset of letters, compute the total factorial divided by the product of factorials for each repeated letter's frequency. For

\text{SEES}

, there are 4 total letters with 2 'S's and 2 'E's, resulting in

\frac{4!}{2!2!} = \frac{24}{4} = 6

.

Adım Adım Çözüm

1

Count the total number of letters and identify frequencies of repeated letters.

The word

\text{SEES}

contains 4 letters in total: two 'S's and two 'E's.

To apply the distinct permutations formula, we need the total count of elements and the counts for each repeated identical element.

2

Calculate the number of distinct arrangements using the formula

\frac{n!}{n_1! n_2! \dots n_k!}

.

\frac{4!}{2! \cdot 2!} = \frac{24}{2 \cdot 2} = 6.

Dividing by

2! \cdot 2!

eliminates duplicate counts arising from swapping indistinguishable identical letters.

Anahtar Kavram

Permutations with Repetition

Soru 215Soru →

What is the numerical value of the expression $\sqrt[3]{7 + 5\sqrt{2}} - \sqrt[3]{5\sqrt{2} - 7}$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The numerical value of the expression is 2.

The value of the expression is 2. This can be demonstrated either by setting the expression equal to

x

, cubing both sides to construct the cubic equation

x^3 + 3x - 14 = 0

, and factoring out the real root

x = 2

, or by recognizing that

(1 + \sqrt{2})^3 = 7 + 5\sqrt{2}

and

(\sqrt{2} - 1)^3 = 5\sqrt{2} - 7

, which simplifies the expression directly to

(1 + \sqrt{2}) - (\sqrt{2} - 1) = 2

.

Adım Adım Çözüm

1

Define variables for the two cubic terms and write the target expression as a difference.

Let

u = \sqrt[3]{7 + 5\sqrt{2}}

and

v = \sqrt[3]{5\sqrt{2} - 7}

, so the target value is

x = u - v

.

Grouping nested radical terms simplifies the algebraic manipulation.

2

Cube both sides of

x = u - v

using the algebraic identity

(u - v)^3 = u^3 - v^3 - 3uv(u - v)

.

x^3 = u^3 - v^3 - 3uv \cdot x

.

Cubing eliminates the outer radical signs on the cubed terms.

3

Evaluate

u^3 - v^3

and the product

uv

.

u^3 - v^3 = (7 + 5\sqrt{2}) - (5\sqrt{2} - 7) = 14

, and

uv = \sqrt[3]{(5\sqrt{2}+7)(5\sqrt{2}-7)} = \sqrt[3]{50 - 49} = 1

.

Using the difference of squares under the cube root simplifies the product term to 1.

4

Substitute the evaluated terms into the cubic equation and solve for the real root

x

.

x^3 = 14 - 3x \implies x^3 + 3x - 14 = 0 \implies (x - 2)(x^2 + 2x + 7) = 0 \implies x = 2

.

The quadratic factor

x^2 + 2x + 7

has negative discriminant (

4 - 28 = -24

), leaving

x = 2

as the unique real solution.

Anahtar Kavram

Simplifying nested radicals using cubic algebraic identities and binomial expansions

Soru 216Soru →

A delivery drone flies in a straight line from Hub A to Hub B against a constant headwind of $10$ miles per hour, taking $4$ hours to complete the flight. On the return flight from Hub B to Hub A along the exact same path, the wind direction reverses to become a tailwind, and its speed increases by $50$ percent. If the return flight takes $2$ hours and the drone maintains a constant airspeed in still air throughout both flights, what was the drone's average speed, in miles per hour, for the entire round trip?

16\frac{2}{3}

20

33\frac{1}{3}

35

37\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $33\frac{1}{3}$

Cevap

The drone's average speed for the entire round trip is

33\frac{1}{3}

miles per hour.

The drone's still-air speed

v

is determined by equating the outbound and return distance equations:

4(v - 10) = 2(v + 15)

, giving

v = 35

miles per hour. The outbound ground speed is

25

miles per hour over

100

miles, and the return ground speed is

50

miles per hour over

100

miles. Dividing the total round-trip distance of

200

miles by the total time of

6

hours yields an average speed of

\frac{200}{6} = 33\frac{1}{3}

miles per hour.

Adım Adım Çözüm

1

Express the effective ground speeds and distance for both legs of the trip in terms of the drone's still-air speed

v

.

Outbound headwind speed =

10

mph, ground speed =

v - 10

mph, outbound distance

D = 4(v - 10)

. Return tailwind speed =

10 \times (1 + 0.50) = 15

mph, return ground speed =

v + 15

mph, return distance

D = 2(v + 15)

.

Distance equals speed multiplied by time, and wind speeds adjust the effective ground speed depending on direction.

2

Equate the distance expressions to solve for the still-air speed

v

.

4(v - 10) = 2(v + 15) \implies 4v - 40 = 2v + 30 \implies 2v = 70 \implies v = 35

mph.

The distance from Hub A to Hub B is identical in both directions.

3

Calculate the one-way distance

D

and total round-trip distance.

One-way distance

D = 4(35 - 10) = 100

miles. Total round-trip distance

= 100 + 100 = 200

miles.

Average speed requires total distance covered across both legs.

4

Calculate the average speed for the entire round trip using

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

.

Total time

= 4 + 2 = 6

hours. Average speed

= \frac{200}{6} = \frac{100}{3} = 33\frac{1}{3}

mph.

Average speed over multiple legs is defined as total distance divided by total elapsed time.

Anahtar Kavram

Average Speed for Multi-Leg Journeys with Wind Vector Effects

Tahmini Süre:2m 30s

Soru 217Soru →

A craft brewery produces two specialty beverages, Batch A and Batch B, using two primary ingredients: hops and malt. To produce 1 barrel of Batch A, the facility requires 4 kilograms of hops and 10 kilograms of malt. To produce 1 barrel of Batch B, the facility requires 6 kilograms of hops and 15 kilograms of malt. Let $a$ represent the number of barrels of Batch A produced and $b$ represent the number of barrels of Batch B produced in a week, where $a > 0$ and $b > 0$ . The total mass of hops used is $H$ kilograms, and the total mass of malt used is $M$ kilograms. Which of the following statements must be true for any valid production quantities of $a$ and $b$ ? Select all that apply.

Geçerli olan tümünü seçin

2M = 5H

M > H

3H - 2M = 0

The ratio of total malt used to total hops used,

\frac{M}{H}

, depends on the specific values of

a

and

b

.

If

H = 120

, then the number of barrels of Batch A produced (

a

) must be equal to 15.

Cevabı ve açıklamayı göster

Cevap: $2M = 5H$ ; $M > H$

Cevap

The statements

2M = 5H

and

M > H

must both be true.

Expressing

H = 4a + 6b = 2(2a + 3b)

and

M = 10a + 15b = 5(2a + 3b)

reveals that

M = 2.5H

, or equivalently

2M = 5H

. Because

H

is strictly positive for positive values of

a

and

b

,

M = 2.5H

also guarantees that

M > H

.

Adım Adım Çözüm

1

Set up algebraic equations for total hops (

H

) and total malt (

M

) in terms of

a

and

b

.

H = 4a + 6b

and

M = 10a + 15b

.

Each barrel of Batch A uses 4 kg hops and 10 kg malt, while each barrel of Batch B uses 6 kg hops and 15 kg malt.

2

Factor common numerical terms from both algebraic expressions.

H = 2(2a + 3b)

and

M = 5(2a + 3b)

.

Factoring isolates the common linear factor

(2a + 3b)

present in both quantities.

3

Calculate the ratio

\frac{M}{H}

and clear fractions to find the invariant linear equation.

\frac{M}{H} = \frac{5(2a + 3b)}{2(2a + 3b)} = \frac{5}{2} \implies 2M = 5H.

Since

a > 0

and

b > 0

,

(2a + 3b) \neq 0

, so the variable terms cancel completely.

4

Evaluate the inequality relationship between

M

and

H

.

M = 2.5H \implies M - H = 1.5H > 0 \implies M > H

.

Because

a, b > 0

,

H > 0

, making

2.5H

strictly greater than

H

.

Anahtar Kavram

Linear Equation Modeling and Proportional Invariants

Soru 218Soru →

If $x$ is a real number such that $x^2 - 5x + 1 = 0$ , what is the value of $x^2 + \frac{1}{x^2}$ ?

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

The value of

x^2 + \frac{1}{x^2}

is 23.

Dividing the quadratic equation

x^2 - 5x + 1 = 0

by

x

yields

x + \frac{1}{x} = 5

. Squaring both sides of this identity gives

\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 25

. Subtracting 2 from both sides results in

x^2 + \frac{1}{x^2} = 23

.

Adım Adım Çözüm

1

Divide the given quadratic equation

x^2 - 5x + 1 = 0

by

x

x - 5 + \frac{1}{x} = 0 \implies x + \frac{1}{x} = 5

Since

x \neq 0

, dividing by

x

isolates the sum of

x

and its reciprocal.

2

Square both sides of the expression

x + \frac{1}{x} = 5

\left(x + \frac{1}{x}\right)^2 = 25 \implies x^2 + 2 + \frac{1}{x^2} = 25

Expanding the binomial square produces the required quadratic sum along with a constant cross-term.

3

Isolate

x^2 + \frac{1}{x^2}

by subtracting 2 from 25

x^2 + \frac{1}{x^2} = 23

Subtracting the constant cross-term yields the exact requested numeric value.

Anahtar Kavram

Algebraic transformation of quadratic equations into reciprocal power sums

Soru 219Soru →

If $n$ is a positive integer such that $140n$ is a perfect square and $105n$ is a perfect cube, what is the minimum possible number of positive divisors of $n$ ?

12

27

108

756

15

Cevabı ve açıklamayı göster

Cevap: 108

Cevap

108

To minimize the number of divisors of

n

, express

n = 2^a \cdot 3^b \cdot 5^c \cdot 7^d

. The condition that

140n = 2^{a+2} \cdot 3^b \cdot 5^{c+1} \cdot 7^{d+1}

is a perfect square implies

a

and

b

must be even, while

c

and

d

must be odd. The condition that

105n = 2^a \cdot 3^{b+1} \cdot 5^{c+1} \cdot 7^{d+1}

is a perfect cube requires

a

to be a multiple of 3, and

b, c, d

to leave a remainder of 2 when divided by 3. The minimum non-negative integers satisfying both requirements are

a=0

(since 0 is even and a multiple of 3),

b=2

(even and leaves remainder 2 mod 3),

c=5

(odd and leaves remainder 2 mod 3), and

d=5

(odd and leaves remainder 2 mod 3). The minimum number of positive divisors is

(0+1)(2+1)(5+1)(5+1) = 108

.

Adım Adım Çözüm

1

Find the prime factorizations of 140 and 105

140 = 2^2 \cdot 5 \cdot 7

and

105 = 3 \cdot 5 \cdot 7

Decomposing into prime factors allows analyzing exponent constraints for perfect powers.

2

Express

n

in terms of prime factors

n = 2^a \cdot 3^b \cdot 5^c \cdot 7^d

and determine constraints for

140n

to be a perfect square

140n = 2^{a+2} \cdot 3^b \cdot 5^{c+1} \cdot 7^{d+1}

requires

a

to be even,

b

to be even,

c

to be odd, and

d

to be odd.

All prime factor exponents in a perfect square must be even numbers.

3

Determine constraints for

105n

to be a perfect cube

105n = 2^a \cdot 3^{b+1} \cdot 5^{c+1} \cdot 7^{d+1}

requires

a \equiv 0 \pmod 3

,

b \equiv 2 \pmod 3

,

c \equiv 2 \pmod 3

, and

d \equiv 2 \pmod 3

.

All prime factor exponents in a perfect cube must be multiples of 3.

4

Find the smallest non-negative integers satisfying both sets of constraints for each exponent

a = 0

,

b = 2

,

c = 5

, and

d = 5

For

a

: smallest non-negative even multiple of 3 is 0. For

b

: smallest non-negative even number congruent to

2 \pmod 3

is 2. For

c

and

d

: smallest odd numbers congruent to

2 \pmod 3

are 5.

5

Calculate the total number of positive divisors of

n = 3^2 \cdot 5^5 \cdot 7^5

(2 + 1)(5 + 1)(5 + 1) = 3 \times 6 \times 6 = 108

The divisor counting formula multiplies

(e_i + 1)

for each prime exponent

e_i

.

Anahtar Kavram

Prime factor exponent constraints for perfect powers and the divisor counting formula

Soru 220Soru →

For any positive integer $n$ , let $S_n$ denote the units digit of the sum $2^n + 3^n + 4^n + 7^n$ . What is the remainder when the sum $T = \sum_{n=1}^{102} S_n$ is divided by $9$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The remainder when the sum T is divided by 9 is 1.

Evaluating the units digits of each exponent term reveals that

S_n

follows a repeating 4-term cycle

(6, 8, 2, 4)

with a sum of 20 per cycle. For 102 terms, there are 25 full cycles plus the first two terms (

S_1=6

and

S_2=8

). The total sum is

T = 25 \times 20 + 6 + 8 = 514

. Dividing 514 by 9 yields

514 = 9 \times 57 + 1

, so the remainder is 1.

Adım Adım Çözüm

1

Find the cyclicity of the units digit of each term

2^n, 3^n, 4^n,

and

7^n

.

Units digits repeat in patterns of length 4: for

2^n

(2, 4, 8, 6), for

3^n

(3, 9, 7, 1), for

4^n

(4, 6, 4, 6), and for

7^n

(7, 9, 3, 1).

Units digits of positive integer powers cycle with periodicities that divide 4.

2

Compute

S_n

for

n = 1, 2, 3, 4

and find the sum of one 4-term period.

S_1 = \text{units}(2+3+4+7=16) = 6

,

S_2 = \text{units}(4+9+6+9=28) = 8

,

S_3 = \text{units}(8+7+4+3=22) = 2

,

S_4 = \text{units}(6+1+6+1=14) = 4

. Sum of one period =

6+8+2+4 = 20

.

The sum of the units digits of individual terms determines the units digit of the total expression.

3

Calculate the total sum

T = \sum_{n=1}^{102} S_n

.

Since

102 = 25 \times 4 + 2

, the sequence consists of 25 complete cycles of 4 terms plus the first 2 terms (

S_1=6

and

S_2=8

). Thus,

T = 25 \times 20 + 6 + 8 = 500 + 14 = 514

.

Dividing the total number of terms by the period length gives the number of full cycles and remaining initial terms.

4

Compute the remainder when

T = 514

is divided by 9.

The sum of the digits of 514 is

5 + 1 + 4 = 10

, and

10 \equiv 1 \pmod 9

. Alternatively,

514 = 9 \times 57 + 1

. Thus, the remainder is 1.

A positive integer and the sum of its digits leave the same remainder when divided by 9.

Anahtar Kavram

Units digit cyclicity of exponential terms and modular arithmetic on sequence sums

Tahmini Süre:2m 0s

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