Tüm alıştırma soruları

Soru 861Soru →

A logistics company assigns identification codes to container shipments based on a positive integer $N$ . The number $N$ has exactly $24$ positive factors, and its prime factorization contains only the prime factors $2$ , $3$ , and $7$ . If $N$ is a multiple of $14$ and the number of odd positive factors of $N$ is $6$ , what is the smallest possible value of $N$ ?

504

1176

252

1512

Cevabı ve açıklamayı göster

Cevap: $504$

Cevap

The smallest possible value of

N

is

504

.

The number

504

has prime factorization

2^3 \times 3^2 \times 7^1

. Total positive factors =

(3+1)(2+1)(1+1) = 24

, and odd positive factors =

(2+1)(1+1) = 6

. It is divisible by

14

(

504 = 14 \times 36

) and is the smaller of the two valid numbers (

504

and

1176

).

Adım Adım Çözüm

1

Express

N

in terms of its prime factors.

N = 2^a \times 3^b \times 7^c

, where

a \ge 1, b \ge 1, c \ge 1

as

N

contains prime factors

2, 3,

and

7

, and is a multiple of

14 = 2 \times 7

.

The question specifies that

N

is divisible by

14

and contains prime factors

2

,

3

, and

7

.

2

Formulate equations for total factors and odd factors.

Total positive factors

T(N) = (a + 1)(b + 1)(c + 1) = 24

. Odd positive factors depend only on odd prime powers (

3^b \times 7^c

), so

(b + 1)(c + 1) = 6

.

Odd factors correspond to choosing

2^0 = 1

for the even prime factor.

3

Solve for exponent

a

.

(a + 1) \times 6 = 24 \implies a + 1 = 4 \implies a = 3

.

Dividing total factors by odd factors isolates the term

(a + 1)

.

4

Find possible values for

b

and

c

.

Since

(b + 1)(c + 1) = 6

with

b, c \ge 1

, the possible factor pairs for

(b+1, c+1)

are

(2, 3)

or

(3, 2)

. Case 1:

b=1, c=2 \implies N = 2^3 \times 3^1 \times 7^2 = 1176

. Case 2:

b=2, c=1 \implies N = 2^3 \times 3^2 \times 7^1 = 504

.

Integer factorizations of

6

give two distinct exponent assignments.

5

Determine the minimum value.

Comparing

504

and

1176

, the minimum value is

504

.

Assigning the larger exponent to the smaller prime base (

3

instead of

7

) minimizes the product.

Anahtar Kavram

Factors and Prime Factorization Constraints

Tahmini Süre:2m 0s

Soru 862Soru →

What is the total number of positive even factors of the integer $180$ ?

12

18

6

9

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The total number of positive even factors of

180

is

12

.

The prime factorization of

180

is

2^2 \times 3^2 \times 5^1

. An even factor must be divisible by

2

, so the power of

2

must be either

1

or

2

(

2

choices). The power of

3

can be

0, 1,

or

2

(

3

choices), and the power of

5

can be

0

or

1

(

2

choices). Total even factors =

2 \times 3 \times 2 = 12

.

Adım Adım Çözüm

1

Express

180

in terms of its prime factors.

180 = 2^2 \times 3^2 \times 5^1

Prime factorization allows systematic calculation of factor counts using exponent combinations.

2

Determine the number of valid choices for each prime factor's exponent to form an even factor.

Choices for exponent of

2

:

1

or

2

(

2

choices). Choices for exponent of

3

:

0, 1

, or

2

(

3

choices). Choices for exponent of

5

:

0

or

1

(

2

choices).

An even factor must contain at least one factor of

2

, so the exponent of

2

cannot be

0

.

3

Multiply the counts of choices together.

2 \times 3 \times 2 = 12

By the fundamental principle of counting, multiplying the available choices for each prime factor gives the total number of even factors.

Anahtar Kavram

Counting Even Factors using Prime Factorization

Soru 863Soru →

Under a State Skill Development Mission, a total of $4,500$ candidates were enrolled across three vocational sectors: IT & Electronics, Healthcare & Allied Sciences, and Textile & Apparel. The IT & Electronics sector accounted for $40\%$ of the total enrolled candidates. The ratio of candidates enrolled in Healthcare & Allied Sciences to those in Textile & Apparel was $7 : 8$ . Upon completion of the training, $65\%$ of the enrolled candidates in IT & Electronics successfully obtained certification. The number of certified candidates in Healthcare & Allied Sciences was equal to $80\%$ of the certified candidates in IT & Electronics. If the total number of certified candidates across all three sectors combined was $2,898$ , what percentage of the enrolled candidates in the Textile & Apparel sector successfully obtained certification?

Cevabı ve açıklamayı göster

Cevap: 55

Cevap

The certification percentage of enrolled candidates in the Textile & Apparel sector is

55\%

.

To find the certification percentage for the Textile & Apparel sector, first determine the enrollment figures: IT & Electronics has

1,800

candidates (

40\%

of

4,500

), leaving

2,700

candidates split in a

7 : 8

ratio between Healthcare (

1,260

) and Textile (

1,440

). Next, calculate certified candidates: IT & Electronics has

1,170

certified (

65\%

of

1,800

), and Healthcare has

936

certified (

80\%

of

1,170

). Subtracting these from the total

2,898

certified candidates leaves

792

certified candidates in Textile & Apparel. Finally,

\frac{792}{1,440} \times 100\% = 55\%

.

Adım Adım Çözüm

1

Calculate the number of candidates enrolled in IT & Electronics

Enrolled in IT & Electronics =

0.40 \times 4,500 = 1,800

candidates.

IT & Electronics accounts for

40\%

of the overall

4,500

candidates.

2

Determine enrollment in Healthcare & Allied Sciences and Textile & Apparel

Remaining candidates =

4,500 - 1,800 = 2,700

. Enrolled in Healthcare =

\frac{7}{15} \times 2,700 = 1,260

. Enrolled in Textile =

\frac{8}{15} \times 2,700 = 1,440

.

The remaining

2,700

candidates are distributed between Healthcare and Textile in the ratio

7 : 8

(total

15

parts).

3

Calculate certified candidates in IT & Electronics

Certified in IT & Electronics =

0.65 \times 1,800 = 1,170

candidates.

65\%

of the enrolled IT & Electronics candidates achieved certification.

4

Calculate certified candidates in Healthcare & Allied Sciences

Certified in Healthcare =

0.80 \times 1,170 = 936

candidates.

Healthcare certified candidates equal

80\%

of certified candidates in IT & Electronics.

5

Determine certified candidates in Textile & Apparel

Certified in Textile =

2,898 - (1,170 + 936) = 2,898 - 2,106 = 792

candidates.

Total certified across all sectors is

2,898

.

6

Compute the certification rate for Textile & Apparel

Certification percentage =

\left(\frac{792}{1,440}\right) \times 100\% = 55\%

.

Divide certified Textile candidates by total enrolled Textile candidates and convert to percentage.

Anahtar Kavram

Data extraction, ratio partitioning, and sequential percentage calculation from unstructured text caselets.

Soru 864Soru →

Is the integer $x$ odd?

Statement (I): $x^2 + 3x$ is an even integer.
Statement (II): $x + 5$ is an even integer.

Which of the following options is correct?

Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient.

Neither Statement (I) nor Statement (II) is sufficient.

Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Cevap

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Evaluating Statement (I):

x^2 + 3x = x(x+3)

. The product of any integer

x

and

(x+3)

is always even because one of the two numbers is always even. Thus, Statement (I) provides no specific information about whether

x

is odd or even, making it insufficient. Evaluating Statement (II):

x + 5 = \text{even}

. Subtracting the odd integer

5

from an even integer yields an odd integer, so

x

must be odd. Hence, Statement (II) alone is sufficient to answer the question definitively.

Adım Adım Çözüm

1

Evaluate Statement (I) individually.

Rewrite the expression as

x^2 + 3x = x(x + 3)

. If

x

is even, then

x(x + 3) = \text{even} \times \text{odd} = \text{even}

. If

x

is odd, then

x(x + 3) = \text{odd} \times \text{even} = \text{even}

. Thus,

x^2 + 3x

is always even regardless of whether

x

is odd or even.

Since the statement holds true for all integers

x

, it cannot determine whether

x

is odd. Therefore, Statement (I) alone is NOT sufficient.

2

Evaluate Statement (II) individually.

The statement gives that

x + 5

is an even integer. Since

5

is an odd integer,

\text{Odd} + \text{Odd} = \text{Even}

, which implies

x

must be an odd integer.

This yields a definitive 'Yes' answer to the question 'Is

x

odd?'. Therefore, Statement (II) alone IS sufficient.

3

Conclude the data sufficiency evaluation.

Statement (II) alone is sufficient to answer the question, but Statement (I) alone is not sufficient.

Each statement was evaluated independently first, rendering statement combination unnecessary.

Anahtar Kavram

Data Sufficiency Parity Analysis

Tahmini Süre:1m 30s

Soru 865Soru →

The table below presents the operational performance and energy distribution metrics of four regional power grids during FY 2025–26. Net available energy for distribution is determined by first deducting transmission and storage losses from total generation, and then subtracting auxiliary station consumption.

Regional Grid	Total Energy Generated (GWh)	Transmission & Storage Loss (%)	Auxiliary Station Consumption (GWh)	Commercial Sector Share of Net Available Energy (%)
Northern Grid	12,000	15%	200	40%
Southern Grid	15,000	12%	400	35%
Eastern Grid	10,000	20%	300	50%
Western Grid	18,000	10%	200	25%

Based on the data provided, what is the ratio of the total energy distributed to the commercial sector by the Southern Grid to the combined energy distributed to the commercial sector by the Northern and Western Grids?

14 : 25

97 : 178

154 : 271

21 : 40

Cevabı ve açıklamayı göster

Cevap: 14 : 25

Cevap

The ratio of energy distributed to the commercial sector by the Southern Grid to the combined commercial distribution of the Northern and Western Grids is 14 : 25.

The correct option stating '14 : 25' accurately follows the prescribed two-stage deduction process. For the Southern Grid, subtracting 12% loss (

1,800\text{ GWh}

) from

15,000\text{ GWh}

gives

13,200\text{ GWh}

, and subtracting

400\text{ GWh}

auxiliary consumption leaves

12,800\text{ GWh}

. Applying

35\%

yields

4,480\text{ GWh}

. For the Northern Grid, net energy is

(12,000 - 1,800) - 200 = 10,000\text{ GWh}

, giving

40\% \times 10,000 = 4,000\text{ GWh}

. For the Western Grid, net energy is

(18,000 - 1,800) - 200 = 16,000\text{ GWh}

, giving

25\% \times 16,000 = 4,000\text{ GWh}

. Combining Northern and Western gives

8,000\text{ GWh}

. The ratio

\frac{4,480}{8,000}

simplifies precisely to

14 : 25

.

Adım Adım Çözüm

1

Calculate the net available energy and commercial distribution for the Southern Grid.

Net Available =

(15,000 - 0.12 \times 15,000) - 400 = (15,000 - 1,800) - 400 = 12,800\text{ GWh}

. Commercial Distribution =

35\% \text{ of } 12,800 = 4,480\text{ GWh}

.

Transmission loss must be deducted from total generation first, followed by auxiliary consumption.

2

Calculate the net available energy and commercial distribution for the Northern Grid.

Net Available =

(12,000 - 0.15 \times 12,000) - 200 = (12,000 - 1,800) - 200 = 10,000\text{ GWh}

. Commercial Distribution =

40\% \text{ of } 10,000 = 4,000\text{ GWh}

.

Required to compute the combined baseline for Northern and Western grids.

3

Calculate the net available energy and commercial distribution for the Western Grid.

Net Available =

(18,000 - 0.10 \times 18,000) - 200 = (18,000 - 1,800) - 200 = 16,000\text{ GWh}

. Commercial Distribution =

25\% \text{ of } 16,000 = 4,000\text{ GWh}

.

Required to complete the denominator sum of the target ratio.

4

Compute the target ratio of Southern Grid commercial distribution to Northern + Western Grid commercial distribution.

Combined Northern + Western =

4,000 + 4,000 = 8,000\text{ GWh}

. Ratio =

4,480 : 8,000 = 14 : 25

.

Simplifying

\frac{4,480}{8,000}

by dividing both terms by their greatest common divisor

160

yields

14 : 25

.

Anahtar Kavram

Multi-step tabular data interpretation involving sequential percentage reduction, absolute deductions, and ratio simplification.

Soru 866Soru →

What is the simplified numerical value of the mathematical expression given below when evaluated strictly according to the VBODMAS rule?

16.5 + \left\{ 4.5 \times \left[ 12 - \left( 3.2 + \overline{4.8 - 1.6} \div 0.8 \right) \right] \right\}

38.1

34.5

100.8

43.5

Cevabı ve açıklamayı göster

Cevap: $38.1$

Cevap

The simplified numerical value of the expression is

38.1

.

Following the VBODMAS priority sequence: resolving the vinculum gives

3.2

; evaluating division inside the round bracket yields

3.2 \div 0.8 = 4

; finishing the round bracket yields

3.2 + 4 = 7.2

; evaluating the square bracket gives

12 - 7.2 = 4.8

; evaluating the curly bracket gives

4.5 \times 4.8 = 21.6

; finally, adding

16.5 + 21.6

yields

38.1

.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{4.8 - 1.6} = 3.2

According to the VBODMAS rule, operations under a vinculum take highest priority.

2

Evaluate the division inside the innermost round bracket

3.2 \div 0.8 = 4

Division takes precedence over addition within parentheses.

3

Complete the evaluation inside the round bracket

3.2 + 4 = 7.2

Add the remaining terms within the round bracket.

4

Evaluate the expression inside the square bracket

12 - 7.2 = 4.8

Subtract the result of the round bracket from

12

.

5

Evaluate the multiplication inside the curly bracket

4.5 \times 4.8 = 21.6

Multiply the term outside the square bracket by the simplified square bracket value.

6

Perform the final addition

16.5 + 21.6 = 38.1

Add the initial constant to the simplified curly bracket term.

Anahtar Kavram

Order of Operations (VBODMAS Rule)

Tahmini Süre:1m 30s

Soru 867Soru →

A large water reservoir is equipped with two inlet pipes, Pipe $P$ and Pipe $Q$ , each filling the reservoir at its own constant rate. How many hours will it take to fill the empty reservoir if both pipes operate simultaneously from the start?

Statement (I): Pipe $P$ alone can fill the empty reservoir in $15\text{ hours}$ .
Statement (II): Pipe $Q$ fills the reservoir at a rate that is $50\%$ higher than the filling rate of Pipe $P$ .

Which of the following options correctly evaluates the sufficiency of the statements?

Statement (I) alone is sufficient to answer the question, while Statement (II) alone is not sufficient.

Statement (II) alone is sufficient to answer the question, while Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient to answer the question.

Neither Statement (I) nor Statement (II) is sufficient, and even together they are not sufficient to answer the question.

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Cevabı ve açıklamayı göster

Cevap: Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Cevap

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating Statement (I) alone gives only the individual performance of Pipe P, which is insufficient to determine the combined time. Statement (II) alone provides only a relative ratio between the rates of Pipe P and Pipe Q without any concrete time metric, making it insufficient on its own. When both statements are combined, Statement (I) provides the base rate for Pipe P and Statement (II) allows calculation of Pipe Q's rate, leading to a unique answer of 6 hours for the combined operation.

Adım Adım Çözüm

1

Evaluate Statement (I) alone

Pipe

P

's rate is

\frac{1}{15}

of the reservoir per hour. However, no information is given regarding Pipe

Q

's rate.

Statement (I) alone is insufficient to calculate the combined time.

2

Evaluate Statement (II) alone

Pipe

Q

's rate is

1.5

times Pipe

P

's rate, meaning Rate(

Q

)

= 1.5 \times

Rate(

P

).

Statement (II) alone gives only a relative ratio of work rates, with no numerical time value given, so it is insufficient.

3

Evaluate Statements (I) and (II) together

From Statement (I), Rate(

P

)

= \frac{1}{15}

reservoir/hour. From Statement (II), Rate(

Q

)

= 1.5 \times \frac{1}{15} = \frac{1}{10}

reservoir/hour. Combined rate

= \frac{1}{15} + \frac{1}{10} = \frac{1}{6}

reservoir/hour. Thus, total combined time

= 6\text{ hours}

.

Combining both statements yields a unique and definitive answer.

Anahtar Kavram

Data Sufficiency in Work and Time / Rate Problems

Soru 868Soru →

A teacher wants to divide $84$ students into equal groups such that the number of students in each group is a prime factor of $84$ . What is the sum of all distinct prime numbers that can represent the size of these groups?

12

14

7

42

Cevabı ve açıklamayı göster

Cevap: $12$

Cevap

The sum of all distinct prime group sizes is

12

.

The prime factorization of

84

is

2^2 \times 3 \times 7

. The distinct prime factors are

2

,

3

, and

7

. Summing these distinct prime numbers yields

2 + 3 + 7 = 12

.

Adım Adım Çözüm

1

Find the prime factorization of

84

.

84 = 2^2 \times 3^1 \times 7^1

.

Decomposing

84

into prime factors reveals all prime numbers that divide

84

exactly.

2

Identify the distinct prime factors of

84

.

The distinct prime factors are

2

,

3

, and

7

.

The question specifies distinct prime factors, so each prime base is listed once regardless of its exponent.

3

Calculate the sum of these distinct prime factors.

2 + 3 + 7 = 12

.

Adding the distinct prime factors gives the sum of all possible prime group sizes.

Anahtar Kavram

Prime Factorization and Distinct Prime Factors

Soru 869Soru →

Let $N$ be a positive integer with the prime factorization $N = 2^4 \times 3^2 \times 5^3$ . Which of the following statements regarding the positive factors of $N$ are correct?

Geçerli olan tümünü seçin

The total number of positive factors of

N

that are perfect squares is

12

.

The number of positive factors of

N

that are divisible by

15

is

30

.

The total number of positive even factors of

N

is

50

.

The number of positive odd factors of

N

is

15

.

Cevabı ve açıklamayı göster

Cevap: The total number of positive factors of $N$ that are perfect squares is $12$ .; The number of positive factors of $N$ that are divisible by $15$ is $30$ .

Cevap

The correct statements are that the total number of positive factors of

N

that are perfect squares is

12

, and the number of positive factors of

N

that are divisible by

15

is

30

.

The statement asserting that

N

has

12

factors that are perfect squares is correct because choosing even powers (

0, 2, 4

for

2

;

0, 2

for

3

; and

0, 2

for

5

) yields

3 \times 2 \times 2 = 12

factors. The statement asserting that

30

factors are divisible by

15

is also correct because forcing exponents of

3

and

5

to be at least

1

yields

5 \times 2 \times 3 = 30

factors.

Adım Adım Çözüm

1

Analyze the prime factorization

N = 2^4 \times 3^2 \times 5^3

Any positive factor of

N

is of the form

2^a \times 3^b \times 5^c

, where

0 \le a \le 4

,

0 \le b \le 2

, and

0 \le c \le 3

.

By the Fundamental Theorem of Arithmetic, factors inherit prime bases with constrained exponents.

2

Count the number of factors that are perfect squares

For a perfect square,

a \in \{0, 2, 4\}

(3 choices),

b \in \{0, 2\}

(2 choices), and

c \in \{0, 2\}

(2 choices). Total square factors

= 3 \times 2 \times 2 = 12

.

All prime exponents in a perfect square must be even non-negative integers.

3

Count the number of factors divisible by

15

Since

15 = 3^1 \times 5^1

,

a \in \{0, 1, 2, 3, 4\}

(5 choices),

b \in \{1, 2\}

(2 choices), and

c \in \{1, 2, 3\}

(3 choices). Total factors divisible by

15 = 5 \times 2 \times 3 = 30

.

Divisibility by

15

mandates at least one factor of

3

and at least one factor of

5

.

4

Verify even and odd factor counts to evaluate the remaining claims

Even factors require

a \ge 1

, giving

4 \times 3 \times 4 = 48

even factors. Odd factors require

a = 0

, giving

1 \times 3 \times 4 = 12

odd factors.

Evaluating

a \ge 1

vs

a = 0

confirms that claims specifying

50

even factors and

15

odd factors are incorrect.

Anahtar Kavram

Counting factors with constrained exponents in prime factorizations

Soru 870Soru →

What is the final numerical value of the following expression when evaluated strictly according to the VBODMAS rule?

12.5 \times 4 - \left[ 16 + \left\{ 25 \div \left( 3.5 + \overline{2.8 - 1.3} \right) \right\} \right]

Cevabı ve açıklamayı göster

Cevap: 29

Cevap

The simplified numerical value of the expression is 29.

Applying VBODMAS step-by-step resolves the vinculum first (2.8 - 1.3 = 1.5), then the round brackets (3.5 + 1.5 = 5), curly brackets (25 / 5 = 5), square brackets (16 + 5 = 21), and finally the multiplication and subtraction (50 - 21 = 29).

Adım Adım Çözüm

1

Evaluate the vinculum (bar) operation first

\overline{2.8 - 1.3} = 1.5

According to VBODMAS, the vinculum takes top priority over all other operations.

2

Simplify the expression inside the innermost round brackets

3.5 + 1.5 = 5

Next in hierarchy are the round brackets (parentheses).

3

Simplify the expression inside the curly brackets

25 \div 5 = 5

Evaluate operations inside curly braces next.

4

Simplify the expression inside the square brackets

16 + 5 = 21

Complete the evaluation of the outer square brackets.

5

Perform the multiplication and final subtraction outside the brackets

12.5 \times 4 = 50 \text{ and then } 50 - 21 = 29

Perform multiplication before final subtraction following standard operator precedence.

Anahtar Kavram

VBODMAS Rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Tahmini Süre:1m 30s

Soru 871Soru →

A positive integer $N$ has the prime factorization $N = 2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. It is known that $N$ has exactly $48$ positive factors, $N/2$ has $36$ positive factors, and $N/5$ has $40$ positive factors. What is the total number of positive factors of $N/3$ ?

24

32

16

0

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The total number of positive factors of

N/3

is 24.

By writing the factor formula

(a+1)(b+1)(c+1) = 48

and comparing it with

d(N/2) = a(b+1)(c+1) = 36

and

d(N/5) = (a+1)(b+1)c = 40

, we uniquely determine the exponents

a = 3

,

b = 1

, and

c = 5

. Dividing

N

by

3

changes the prime exponent of

3

to

0

. Applying the factor count formula to

2^3 \times 3^0 \times 5^5

yields

(3+1)(0+1)(5+1) = 24

.

Adım Adım Çözüm

1

Express the total number of factors of

N

using its prime factorization

d(N) = (a+1)(b+1)(c+1) = 48

The number of positive factors of a number

p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is given by

(e_1+1)(e_2+1)\dots(e_k+1)

.

2

Use the factor counts of

N/2

and

N/5

to determine the exponents

a

and

c

a = 3

and

c = 5

Since

d(N/2) = a(b+1)(c+1) = 36

, taking the ratio

d(N/2)/d(N)

yields

a/(a+1) = 36/48 = 3/4

, which gives

a = 3

. Similarly,

d(N/5) = (a+1)(b+1)c = 40

, so

d(N/5)/d(N) = c/(c+1) = 40/48 = 5/6

, giving

c = 5

.

3

Solve for the remaining exponent

b

b = 1

Substituting

a = 3

and

c = 5

into

d(N) = 48

gives

(3+1)(b+1)(5+1) = 48 \implies 24(b+1) = 48 \implies b+1 = 2 \implies b = 1

.

4

Calculate the number of positive factors of

N/3

d(N/3) = (3+1)(0+1)(5+1) = 4 \times 1 \times 6 = 24

Dividing

N

by

3

reduces the exponent of

3

from

b = 1

to

b-1 = 0

, so

N/3 = 2^3 \times 3^0 \times 5^5

.

Anahtar Kavram

Prime Factorization and Number of Factors

Tahmini Süre:2m 0s

Soru 872Soru →

Consider the positive integer $90$ . Which of the following statements regarding the factors and prime factorization of $90$ are correct?

Geçerli olan tümünü seçin

The prime factorization of

90

is

2^1 \times 3^2 \times 5^1

.

The total number of positive factors of

90

is

12

.

The number of positive even factors of

90

is

8

.

The sum of all distinct prime factors of

90

is

15

.

Cevabı ve açıklamayı göster

Cevap: The prime factorization of $90$ is $2^1 \times 3^2 \times 5^1$ .; The total number of positive factors of $90$ is $12$ .

Cevap

The prime factorization of 90 is 2^1 × 3^2 × 5^1, and the total number of positive factors of 90 is 12.

The prime factorization of 90 is indeed 2^1 × 3^2 × 5^1. The total number of positive factors is calculated as (1+1)(2+1)(1+1) = 12, making both statement options correct.

Adım Adım Çözüm

1

Find the prime factorization of 90.

90 = 2 × 45 = 2 × 3^2 × 5 = 2^1 × 3^2 × 5^1.

Decomposing 90 into prime base powers establishes the prime factorization.

2

Calculate the total number of positive factors using exponents.

Total factors = (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12.

Adding 1 to each prime exponent and multiplying gives the total factor count.

3

Verify even factors and prime factor sum.

Even factors = 1 × (2 + 1)(1 + 1) = 6. Distinct prime sum = 2 + 3 + 5 = 10.

Confirming additional properties disproves the incorrect statements.

Anahtar Kavram

Prime factorization and formulas for determining total and specific factor counts of an integer

Soru 873Soru →

The table below details raw material procurement, fabric output, and sales revenues across four regional textile development clusters for FY 2024–25:

Textile Cluster	Raw Cotton Procured ('000 MT)	Fabric Produced ('000 metres)	Export Value (₹ Crores)	Domestic Sales (₹ Crores)
Cluster A	40	320	160	240
Cluster B	50	500	300	200
Cluster C	60	480	240	360
Cluster D	30	240	180	120

What is the absolute difference (in ₹ Crores) between the average domestic sales across all four clusters and the domestic sales of the cluster that achieved the highest fabric production efficiency per unit of raw cotton procured?

₹30 Crores

₹50 Crores

₹130 Crores

₹30,000

Cevabı ve açıklamayı göster

Cevap: ₹30 Crores

Cevap

The absolute difference is ₹30 Crores.

The correct answer is ₹30 Crores. Fabric production efficiency per unit of raw cotton is calculated as Fabric Produced ('000 metres) divided by Raw Cotton Procured ('000 MT). Cluster B yields 500 / 50 = 10 metres/MT, which is the highest among all clusters (others yield 8 metres/MT). Cluster B's domestic sales are ₹200 Crores. The average domestic sales across all clusters is (240 + 200 + 360 + 120) / 4 = ₹230 Crores. The absolute difference between ₹230 Crores and ₹200 Crores is ₹30 Crores.

Adım Adım Çözüm

1

Calculate fabric production efficiency per unit of raw cotton procured for each cluster

Cluster A: 320 / 40 = 8 metres/MT; Cluster B: 500 / 50 = 10 metres/MT; Cluster C: 480 / 60 = 8 metres/MT; Cluster D: 240 / 30 = 8 metres/MT.

To identify which cluster achieved the highest fabric production per unit of raw cotton procured.

2

Identify the domestic sales of the most efficient cluster

Cluster B achieved the highest efficiency (10 metres/MT). Its domestic sales volume is ₹200 Crores.

This provides the targeted cluster's metric for comparison.

3

Calculate the average domestic sales across all four clusters

Total Domestic Sales = 240 + 200 + 360 + 120 = ₹920 Crores. Average Domestic Sales = 920 / 4 = ₹230 Crores.

To determine the benchmark average domestic sales across the dataset.

4

Compute the absolute difference

| ₹230 Crores - ₹200 Crores | = ₹30 Crores.

To answer the final question requirement.

Anahtar Kavram

Data grid ratio comparison and benchmark deviation analysis

Soru 874Soru →

A rectangular courtyard measuring $105\text{ cm}$ in length and $135\text{ cm}$ in width is to be completely paved with identical square tiles without cutting any tile. What is the maximum possible side length of each square tile?

15\text{ cm}

5\text{ cm}

45\text{ cm}

945\text{ cm}

Cevabı ve açıklamayı göster

Cevap: $15\text{ cm}$

Cevap

The maximum possible side length of each square tile is

15\text{ cm}

.

The maximum side length of a square tile that fits exact integer counts along both length and width is given by

\text{HCF}(105, 135)

. By prime factorization,

105 = 3 \times 5 \times 7

and

135 = 3^3 \times 5

. The common prime factors are

3

and

5

, each with lowest power

1

. Therefore,

\text{HCF} = 3 \times 5 = 15\text{ cm}

.

Adım Adım Çözüm

1

Identify the required mathematical operation

To find the maximum side length of identical square tiles that fit whole dimensions of

105\text{ cm}

and

135\text{ cm}

, find the Highest Common Factor (HCF) of

105

and

135

.

The side of each square tile must divide both the length and the width completely, and to maximize tile size, we need the largest such divisor.

2

Perform prime factorization of both numbers

105 = 3 \times 5 \times 7

135 = 3^3 \times 5 = 3 \times 3 \times 3 \times 5

Expressing numbers as products of prime factors allows easy identification of common factors.

3

Calculate the HCF using prime powers

\text{HCF}(105, 135) = 3^1 \times 5^1 = 15

HCF is obtained by taking the product of the smallest powers of all common prime factors.

Anahtar Kavram

Application of HCF and Prime Factorization in Spatial Tiling Problems

Tahmini Süre:45s

Soru 875Soru →

Let $N = 2^5 \times 3^4 \times 5^2$ . How many positive integer factors of $N$ are divisible by $12$ but not divisible by $72$ ?

Cevabı ve açıklamayı göster

Cevap: 21

Cevap

21

The correct answer is 21. By analyzing the exponent bounds of

2^a \times 3^b \times 5^c

, there are 48 factors of

N

divisible by

12

(

a \ge 2, b \ge 1

). Among these, 27 factors are also divisible by

72

(

a \ge 3, b \ge 2

). Subtracting the two counts gives

48 - 27 = 21

factors divisible by 12 but not by 72.

Adım Adım Çözüm

1

Represent the general form of a factor of

N

Every positive factor of

N = 2^5 \times 3^4 \times 5^2

can be uniquely written as

f = 2^a \times 3^b \times 5^c

, where

a \in \{0, 1, 2, 3, 4, 5\}

,

b \in \{0, 1, 2, 3, 4\}

, and

c \in \{0, 1, 2\}

.

By the Fundamental Theorem of Arithmetic, any factor of a number in prime-factored form takes exponents bounded by the original number's exponents.

2

Identify exponent constraints for divisibility by 12

Since

12 = 2^2 \times 3^1

, for

f

to be divisible by

12

, we must have

a \ge 2

and

b \ge 1

.

Divisibility requires prime factor exponents in the factor to be at least as large as those in the divisor.

3

Calculate the total number of factors divisible by 12

Number of choices for

a

:

5 - 2 + 1 = 4

(values

2, 3, 4, 5

). Number of choices for

b

:

4 - 1 + 1 = 4

(values

1, 2, 3, 4

). Number of choices for

c

:

2 - 0 + 1 = 3

(values

0, 1, 2

). Total count =

4 \times 4 \times 3 = 48

.

By the multiplication principle of counting, independent choices for each exponent are multiplied.

4

Identify exponent constraints and count factors divisible by 72

Since

72 = 2^3 \times 3^2

, a factor is divisible by

72

if

a \ge 3

and

b \ge 2

. Number of choices:

a \in \{3, 4, 5\}

(3 choices),

b \in \{2, 3, 4\}

(3 choices),

c \in \{0, 1, 2\}

(3 choices). Total count =

3 \times 3 \times 3 = 27

.

Every factor of

N

that is divisible by 72 is inherently also a factor divisible by 12, forming a strict subset.

5

Apply the principle of subtraction to find the required count

Required count = (Factors divisible by 12) - (Factors divisible by 72) =

48 - 27 = 21

.

The set of factors divisible by 12 but not by 72 is the set difference between factors divisible by 12 and factors divisible by 72.

Anahtar Kavram

Counting Factors with Divisibility Constraints using Prime Factorization

Tahmini Süre:2m 0s

Soru 876Soru →

Calculate the exact numerical value of the given mathematical expression using the VBODMAS rule:

\frac{3}{5} \text{ of } 250 - \left[ 42 + \left\{ 24 \div \left( 7.5 - \overline{4.2 + 0.3} \right) \right\} \right]

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

100

Evaluating strictly according to VBODMAS order of operations yields 100: Vinculum gives 4.5, round brackets give 3, division in curly brackets gives 8, addition in square brackets gives 50, 'of' operation gives 150, and 150 - 50 = 100.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar line)

4.2 + 0.3 = 4.5

According to the VBODMAS rule, the vinculum takes precedence over standard brackets and operations.

2

Simplify the innermost round brackets

7.5 - 4.5 = 3

Operations inside round parentheses ( ) are resolved next.

3

Evaluate the expression inside the curly brackets

24 / 3 = 8

Perform the division operation inside curly braces { }.

4

Evaluate the expression inside the square brackets

42 + 8 = 50

Perform addition within the outer square brackets [ ].

5

Calculate the 'of' operation

(3/5) * 250 = 150

The 'of' operation takes priority over standard addition and subtraction outside brackets.

6

Perform final subtraction

150 - 50 = 100

Subtract the total evaluated bracketed quantity from the 'of' calculation result.

Anahtar Kavram

Hierarchical priority of operations in VBODMAS involving vinculum, nested brackets, and fractional 'of' operations

Soru 877Soru →

A positive integer $N$ has exactly $15$ positive factors and is divisible by $6$ . If $N$ has exactly two distinct prime factors, what is the sum of all possible values of $N$ that are less than $500$ ?

144

324

468

540

Cevabı ve açıklamayı göster

Cevap: 468

Cevap

The sum of all possible values of N less than 500 is 468.

For a number

N = p^a \times q^b

to have 15 factors,

(a+1)(b+1) = 15

. The factor pairs of 15 for two prime factors are

(5, 3)

and

(3, 5)

, which correspond to exponent pairs

(4, 2)

and

(2, 4)

. Since

N

is divisible by 6, its prime factors are 2 and 3. Computing both cases gives

2^4 \times 3^2 = 144

and

2^2 \times 3^4 = 324

. Both are less than 500, making their sum

144 + 324 = 468

.

Adım Adım Çözüm

1

Determine the prime factors of N

N has prime factors 2 and 3

Since N is divisible by 6, it must have at least 2 and 3 as prime factors. The problem states N has exactly two distinct prime factors, so its prime factorization is of the form

N = 2^a \times 3^b

.

2

Apply the total factors formula

(a + 1)(b + 1) = 15

The total number of positive factors of

N = p^a \times q^b

is given by

(a + 1)(b + 1) = 15

.

3

Find valid non-negative integer pairs (a, b)

Either

(a=4, b=2)

or

(a=2, b=4)

Since 15 factors into

5 \times 3

or

3 \times 5

(note that

15 \times 1

would imply only one prime factor, which contradicts having two distinct prime factors), the possible exponent pairs are

(4, 2)

and

(2, 4)

.

4

Calculate the values of N and check bounds

N_1 = 2^4 \times 3^2 = 144

and

N_2 = 2^2 \times 3^4 = 324

Both

144 < 500

and

324 < 500

meet all conditions.

5

Sum the valid values of N

144 + 324 = 468

Adding the two valid integers yields the total sum.

Anahtar Kavram

Factors and Prime Factorization Exponent Rule

Soru 878Soru →

Three church bells toll together at intervals of $9$ minutes, $12$ minutes, and $15$ minutes respectively. If they all toll together at $8:00\text{ AM}$ , at what time will they next toll together?

8:03 AM

8:36 AM

10:00 AM

11:00 AM

Cevabı ve açıklamayı göster

Cevap: 11:00 AM

Cevap

11:00 AM

The bells will toll together after a duration equal to the Least Common Multiple (LCM) of their individual tolling intervals. The prime factorizations are

9 = 3^2

,

12 = 2^2 \times 3

, and

15 = 3 \times 5

. Taking the highest power of each prime factor yields

\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 180

minutes, which is equal to

3

hours. Adding

3

hours to

8:00\text{ AM}

gives

11:00\text{ AM}

.

Adım Adım Çözüm

1

Find prime factorizations of each interval

9 = 3^2

,

12 = 2^2 \times 3

, and

15 = 3 \times 5

To compute the Least Common Multiple (LCM), express each number in prime factor form.

2

Calculate the LCM

\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\text{ minutes}

Take the highest power of each prime factor involved.

3

Convert minutes to hours and add to the initial time

180\text{ minutes} = 3\text{ hours}

;

8:00\text{ AM} + 3\text{ hours} = 11:00\text{ AM}

Determine the exact time the bells will next chime together.

Anahtar Kavram

Application of Least Common Multiple (LCM) in Simultaneous Periodic Events

Soru 879Soru →

What is the simplified numerical value of the following mathematical expression when evaluated strictly according to the VBODMAS rule?

40\% \text{ of } 150 - \left[ 16 + \left\{ 20 \div \left( 8.5 - \overline{3.1 + 1.4} \right) \right\} \right]

39

51

49

40

Cevabı ve açıklamayı göster

Cevap: 39

Cevap

39

Evaluating strictly by VBODMAS priority: first the vinculum yields 4.5, then the round bracket gives 8.5 - 4.5 = 4, then division yields 20 ÷ 4 = 5, followed by square bracket addition 16 + 5 = 21. Finally, subtracting 21 from 40% of 150 (which is 60) gives 60 - 21 = 39.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{3.1 + 1.4} = 4.5

The vinculum has the highest priority and must be evaluated first.

2

Evaluate the terms inside the round brackets

8.5 - 4.5 = 4

The next priority is resolving the innermost parentheses.

3

Perform division inside the curly brackets

20 \div 4 = 5

Inside curly brackets, division takes precedence over addition.

4

Evaluate the terms inside the square brackets

16 + 5 = 21

Resolve the outer square bracket by adding the terms.

5

Calculate the percentage term

40\% \text{ of } 150 = \frac{40}{100} \times 150 = 60

'Of' represents multiplication in percentage operations.

6

Perform final subtraction

60 - 21 = 39

Subtract the total bracket value from the percentage calculation.

Anahtar Kavram

VBODMAS Rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Soru 880Soru →

Consider the positive integer $N = 720$ , which has a prime factorization of $2^4 \times 3^2 \times 5^1$ . Which of the following statements regarding the positive factors of $N$ are correct?

Geçerli olan tümünü seçin

The number of positive factors of

N

that are perfect squares is

6

.

The sum of all positive even factors of

N

is

2340

.

The number of positive factors of

N

that are multiples of

15

is

6

.

The product of all positive factors of

N

is equal to

720^{15}

.

Cevabı ve açıklamayı göster

Cevap: The number of positive factors of $N$ that are perfect squares is $6$ .; The sum of all positive even factors of $N$ is $2340$ .; The product of all positive factors of $N$ is equal to $720^{15}$ .

Cevap

The correct statements are those asserting that

N

has 6 perfect square factors, that the sum of all positive even factors of

N

is 2340, and that the product of all positive factors of

N

is

720^{15}

.

The statements confirming 6 perfect square factors, a sum of 2340 for even factors, and a factor product of

720^{15}

are all mathematically correct applications of prime factorization principles.

Adım Adım Çözüm

1

Determine perfect square factors of

N = 2^4 \times 3^2 \times 5^1

For a factor

2^a \times 3^b \times 5^c

to be a square, exponents

a, b, c

must be even. Possible values:

a \in \{0, 2, 4\}

(3 options),

b \in \{0, 2\}

(2 options),

c \in \{0\}

(1 option). Number of square factors

= 3 \times 2 \times 1 = 6

.

Perfect squares require all prime factors to have even exponents.

2

Calculate the sum of all positive even factors

Sum of all factors

= (2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2)(5^0+5^1) = 31 \times 13 \times 6 = 2418

. Sum of odd factors (only

2^0

)

= 1 \times 13 \times 6 = 78

. Sum of even factors

= 2418 - 78 = 2340

.

Even factors are obtained by subtracting odd factor sum from total factor sum.

3

Count factors divisible by 15

Since

15 = 3^1 \times 5^1

, factors of 720 divisible by 15 require

a \in \{0, 1, 2, 3, 4\}

(5 choices),

b \in \{1, 2\}

(2 choices), and

c \in \{1\}

(1 choice). Total factors

= 5 \times 2 \times 1 = 10

.

Divisibility by 15 requires at least power 1 for both prime factors 3 and 5.

4

Compute the product of all positive factors

Total factors

T = (4+1)(2+1)(1+1) = 30

. Product of factors

= N^{T/2} = 720^{30/2} = 720^{15}

.

Factors pair up such that

f_i \times f_{T+1-i} = N

, giving

T/2

pairs.

Anahtar Kavram

Properties of positive factors derived from prime factorization: square factor counts, even factor sums, constrained divisibility factor counts, and total factor product formula.

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