Basic Numeracy Alıştırma Soruları — State PSC Exam — Sayfa 3

Evaluate the following mathematical expression using the standard BODMAS (VBODMAS) rule:

40\% \text{ of } 250 - \left[ 15 + \left\{ 45 \div \left( 2 + \frac{1}{4} \text{ of } \left( 18 - \overline{8 - 2} \right) \right) \right\} \right]

What is the final simplified value of the expression?

76

88

94

24

Cevabı ve açıklamayı göster

Cevap: 76

Cevap

The simplified value of the expression is 76.

Following the strict hierarchical order of operations (VBODMAS), resolving the bar vinculum first gives 6, then the inner parenthesis yields 12. Taking 1/4 of 12 gives 3, and adding 2 yields 5. Dividing 45 by 5 yields 9, and adding 15 yields 24 inside the main bracket. Subtracting 24 from 40% of 250 (which equals 100) gives the correct answer of 76.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum bar:

\overline{8 - 2}

.

\overline{8 - 2} = 6

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

2

Evaluate the innermost round bracket:

(18 - 6)

.

18 - 6 = 12

Resolve terms within the round parentheses.

3

Evaluate the 'of' operation inside the round bracket context:

\frac{1}{4} \text{ of } 12

.

\frac{1}{4} \times 12 = 3

'Of' precedes basic addition and division.

4

Complete the addition inside the innermost bracket:

(2 + 3)

.

2 + 3 = 5

Finalize inner parentheses.

5

Evaluate the division inside the curly brackets:

\{45 \div 5\}

.

45 \div 5 = 9

Perform division inside curly brackets.

6

Evaluate the addition inside the square brackets:

[15 + 9]

.

15 + 9 = 24

Resolve the square brackets.

7

Evaluate the percentage term:

40\% \text{ of } 250

.

\frac{40}{100} \times 250 = 100

Calculate the percentage value independently.

8

Perform the final subtraction:

100 - 24

.

100 - 24 = 76

Subtract the bracket result from the percentage value.

Anahtar Kavram

BODMAS / VBODMAS Order of Operations

Alternatif Yöntem

Calculate the percentage term 40% of 250 = 100 first, then systematically simplify the bracket from innermost (vinculum) to outermost to subtract from 100.

Tahmini Süre:1m 15s

Soru 42Soru →

What is the Highest Common Factor (HCF) of the fractions $\frac{8}{15}$ and $\frac{14}{25}$ ?

\frac{2}{75}

\frac{56}{5}

\frac{2}{5}

\frac{56}{75}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{75}$

Cevap

The Highest Common Factor (HCF) is

\frac{2}{75}

.

The correct answer is found by taking the Highest Common Factor (HCF) of the numerators (

8

and

14

), which is

2

, and dividing it by the Least Common Multiple (LCM) of the denominators (

15

and

25

), which is

75

. This yields exactly

\frac{2}{75}

.

Adım Adım Çözüm

1

Identify the numerators and denominators of the given fractions.

The numerators are

8

and

14

. The denominators are

15

and

25

.

The formula for the HCF of fractions requires processing numerators and denominators separately.

2

Calculate the Highest Common Factor (HCF) of the numerators.

The HCF of

8

and

14

is

2

.

According to the formula, the numerator of the final answer must be the HCF of the original numerators.

3

Calculate the Least Common Multiple (LCM) of the denominators.

The LCM of

15

and

25

is

75

.

According to the formula, the denominator of the final answer must be the LCM of the original denominators.

4

Combine the results into a single fraction.

The resulting fraction is

\frac{2}{75}

.

The HCF of fractions is defined as

\frac{\text{HCF of numerators}}{\text{LCM of denominators}}

.

Anahtar Kavram

Highest Common Factor (HCF) of Fractions

Soru 43Soru →

What is the remainder when the mathematical expression $14 \times 47^{55} - 63$ is divided by $48$ ?

29

16

19

1

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

The correct remainder is 19.

The correct answer is found by applying modular arithmetic properties. First, simplify the base:

47 \equiv -1 \pmod{48}

. Raising this to an odd power gives

47^{55} \equiv (-1)^{55} = -1 \pmod{48}

. Substituting this back into the expression yields

14 \times (-1) - 63

. We can also simplify

63 \pmod{48}

to

15

. The expression then becomes

-14 - 15 = -29 \pmod{48}

. Because standard remainders cannot be negative, we add the divisor (48) to find the true positive remainder:

-29 + 48 = 19

.

Adım Adım Çözüm

1

Simplify the base of the exponent modulo 48.

47 \equiv -1 \pmod{48}

Using negative remainders makes calculating large powers manageable.

2

Evaluate the exponentiated term modulo 48.

(-1)^{55} = -1

An odd power of -1 remains -1.

3

Simplify the constant term modulo 48.

63 \equiv 15 \pmod{48}

To keep all terms within the modulo base before finalizing the expression.

4

Substitute the simplified values back into the expression.

14 \times (-1) - 15 = -14 - 15 = -29

To find the overall remainder of the expression, which currently results in a negative value.

5

Convert the negative remainder to a valid positive remainder.

-29 + 48 = 19

Remainders must be strictly positive integers less than the divisor.

Anahtar Kavram

Divisibility Rules and Remainder Theorem

Soru 44Soru →

Arrange the following numerical values in ascending order (from smallest to largest).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is

\frac{2}{5}

,

0.6

,

\frac{7}{10}

, and

0.75

.

When converted to decimals, the values are

0.40

,

0.60

,

0.70

, and

0.75

. Arranging these from smallest to largest yields the correct sequence:

\frac{2}{5}

,

0.6

,

\frac{7}{10}

,

0.75

.

Adım Adım Çözüm

1

Convert the fractions into decimal format to establish a common baseline for comparison.

\frac{2}{5} = 0.4

and

\frac{7}{10} = 0.7

.

Decimals are often much easier to compare directly digit by digit than mixed formats.

2

Append trailing zeros to the decimal values to ensure they all have the same number of decimal places (two places, matching 0.75).

The values to compare are

0.40

,

0.60

,

0.70

, and

0.75

.

Aligning decimal places prevents magnitude confusion and makes visual comparison straightforward.

3

Order the standardized decimals from smallest to largest.

0.40 < 0.60 < 0.70 < 0.75

.

The question explicitly requires an ascending order.

4

Map the ordered decimals back to their original given forms.

\frac{2}{5} < 0.6 < \frac{7}{10} < 0.75

.

The final answer must be presented using the exact numbers provided in the question.

Anahtar Kavram

Converting fractions to decimals to evaluate and compare their magnitudes.

Tahmini Süre:45s

Soru 45Soru →

If the number $11^{2025}$ is decreased by $16^{2025}$ , and the resulting difference is divided by $9$ , what is the remainder?

7

2

4

0

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

To find the remainder of

11^{2025} - 16^{2025}

divided by

9

, we first simplify the bases using modular arithmetic:

11 \equiv 2 \pmod 9

and

16 \equiv -2 \pmod 9

. Next, we evaluate the powers. We know

2^3 = 8 \equiv -1 \pmod 9

. By rewriting the exponent,

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

. For the second term,

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

. Subtracting the two terms yields

-1 - 1 = -2

. Because a remainder must be positive, we add the divisor to the negative result:

-2 + 9 = 7

.

Adım Adım Çözüm

1

Reduce the base numbers modulo 9.

11 \equiv 2 \pmod 9

and

16 \equiv 7 \equiv -2 \pmod 9

.

Reducing the bases simplifies the expression to

2^{2025} - (-2)^{2025} \pmod 9

.

2

Determine the cyclicity pattern for powers of 2 modulo 9.

2^1=2

,

2^2=4

,

2^3=8 \equiv -1 \pmod 9

.

Finding a power that leaves a remainder of

1

or

-1

allows us to easily compute extremely large exponents.

3

Evaluate the first term

2^{2025} \pmod 9

.

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

.

Breaking the exponent

2025

into

3 \times 675

utilizes the known value of

2^3 \pmod 9

.

4

Evaluate the second term

(-2)^{2025} \pmod 9

.

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

.

An odd power preserves the negative sign, allowing substitution of the result from the previous step.

5

Subtract the remainders and map to a positive equivalent.

(-1) - 1 = -2

. To make it positive, add

9

:

-2 + 9 = 7

.

Remainders must be strictly positive and less than the divisor. Adding the divisor to a negative modular result finds its positive equivalent.

Anahtar Kavram

Modular Arithmetic Cyclicity and Negative Remainders

Soru 46Soru →

A decentralized network consists of $13$ primary verification nodes. A smart contract generates a total of $37^{105}$ encrypted tokens that must be distributed equally among these $13$ nodes. The leftover tokens that cannot be distributed equally are sent to a burn address. How many tokens will be sent to the burn address?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The correct remainder when

37^{105}

is divided by

13

is

8

. This is found by reducing the base

37

to

-2

modulo

13

, applying Fermat's Little Theorem to reduce the exponent

105

to

9

modulo

12

, and computing

(-2)^9 \pmod{13} = -18 \equiv 8 \pmod{13}

.

Adım Adım Çözüm

1

Formulate the problem using modular arithmetic.

Evaluate

37^{105} \pmod{13}

.

Since tokens are distributed equally among 13 nodes, the tokens sent to the burn address represent the remainder when the total is divided by 13.

2

Simplify the base of the exponent.

37 = 13 \times 2 + 11

. Using a negative remainder,

11 \equiv -2 \pmod{13}

.

Working with a smaller absolute base value like -2 makes successive exponentiation much easier than working with 11 or 37.

3

Apply Fermat's Little Theorem.

Since 13 is prime,

(-2)^{12} \equiv 1 \pmod{13}

.

This theorem allows us to significantly reduce large exponents by finding their remainder when divided by

p-1

.

4

Reduce the exponent using the theorem.

105 = 12 \times 8 + 9

. Thus,

(-2)^{105} = ((-2)^{12})^8 \times (-2)^9 \equiv 1^8 \times (-2)^9 \equiv (-2)^9 \pmod{13}

.

The multiples of 12 in the exponent evaluate to 1 modulo 13, leaving only the remainder of the exponent.

5

Calculate the final reduced power.

(-2)^4 = 16 \equiv 3 \pmod{13}

. Therefore,

(-2)^8 = 3^2 = 9 \pmod{13}

. Finally,

(-2)^9 = 9 \times (-2) = -18 \pmod{13}

.

Breaking the calculation into smaller powers prevents the need to calculate very large numbers directly.

6

Convert the negative result to a positive remainder.

-18 = 13 \times (-2) + 8

. The positive remainder is

8

.

Physical quantities, such as the number of leftover tokens, must be represented by a positive remainder.

Anahtar Kavram

Modular arithmetic, Fermat's Little Theorem, and negative remainders

Soru 47Soru →

A water storage tank is currently filled to $0.65$ of its total capacity. If the tank currently contains $130$ liters of water, what is the total capacity of the tank in liters?

Cevabı ve açıklamayı göster

Cevap: 200

Cevap

The total capacity of the tank is 200 liters.

The correct capacity is found by dividing the current volume of water (130 liters) by the decimal that represents the filled portion (0.65). This calculation,

130 \div 0.65

, yields 200 liters.

Adım Adım Çözüm

1

Identify the relationship between the filled portion and the total capacity.

Let

C

be the total capacity. We establish the equation:

0.65 \times C = 130

.

Translating the word problem into a mathematical equation allows us to solve for the unknown whole amount.

2

Rearrange the equation to solve for the total capacity.

C = \frac{130}{0.65}

Isolating

C

on one side of the equation gives us the expression needed to find the total capacity.

3

Perform the division by clearing the decimal in the denominator.

C = \frac{13000}{65} = 200

Multiplying the numerator and denominator by 100 eliminates the decimal point, making the division straightforward.

Anahtar Kavram

Calculating the total amount when a specific decimal fraction of that amount is known.

Soru 48Soru →

Evaluate the following mathematical expression:

1.2 + 2.4 \div \frac{4}{5} \times 0.5

What is the correct resulting value?

7.2

2.7

2.25

9.0

Cevabı ve açıklamayı göster

Cevap: 2.7

Cevap

2.7

The correct sequence of operations is to first standardize the numbers (converting

\frac{4}{5}

to

0.8

), perform the division (

2.4 \div 0.8 = 3

), perform the multiplication (

3 \times 0.5 = 1.5

), and finally add the result to the initial value (

1.2 + 1.5 = 2.7

).

Adım Adım Çözüm

1

Convert the fraction to a decimal to make all terms uniform.

\frac{4}{5} = 0.8

. The expression becomes

1.2 + 2.4 \div 0.8 \times 0.5

.

Working entirely in decimals (or entirely in fractions) simplifies the arithmetic process.

2

Apply the order of operations (BODMAS/PEMDAS) by resolving division and multiplication from left to right. First, divide.

2.4 \div 0.8 = 3

. The expression becomes

1.2 + 3 \times 0.5

.

Division and multiplication take precedence over addition, and must be performed sequentially from left to right.

3

Perform the multiplication step.

3 \times 0.5 = 1.5

. The expression becomes

1.2 + 1.5

.

Multiplication is the next operation in the left-to-right sequence.

4

Perform the final addition.

1.2 + 1.5 = 2.7

.

Addition is the last operation remaining in the expression.

Anahtar Kavram

Order of Operations (BODMAS) with Mixed Number Formats

Soru 49Soru →

What is the remainder when $3^{53}$ is divided by $7$ ?

5

2

1

4

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

The remainder when

3^{53}

is divided by

7

is

5

.

By applying modular arithmetic rules,

3^3 = 27 \equiv -1 \pmod{7}

. Expanding

3^{53}

gives

(3^3)^{17} \times 3^2 \equiv (-1)^{17} \times 2 = -2 \pmod{7}

. Adding

7

to the negative remainder yields the standard positive remainder of

5

.

Adım Adım Çözüm

1

Express the base

3

in terms of powers close to a multiple of

7

3^3 = 27 = 7 \times 4 - 1 \equiv -1 \pmod{7}

Finding a power that yields

\pm 1 \pmod{7}

simplifies modular exponentiation.

2

Rewrite

3^{53}

using

3^3

3^{53} = (3^3)^{17} \times 3^2

Break down the exponent

53

as

3 \times 17 + 2

using laws of indices.

3

Evaluate the expression modulo

7

3^{53} \equiv (-1)^{17} \times 9 \equiv -1 \times 2 = -2 \pmod{7}

Since

(-1)

raised to an odd power is

-1

, and

9 \equiv 2 \pmod{7}

.

4

Convert negative remainder to positive equivalent

-2 + 7 = 5

Remainders in standard arithmetic must be non-negative integers in the range

[0, \text{divisor}-1]

.

Anahtar Kavram

Remainder Theorem and Modular Arithmetic

Soru 50Soru →

Consider the fractions $\frac{12}{25}$ , $\frac{18}{35}$ , and $\frac{24}{55}$ . Which of the following statements are mathematically correct regarding their Highest Common Factor (HCF) and Least Common Multiple (LCM)?

Geçerli olan tümünü seçin

The HCF of the given fractions is

\frac{6}{1925}

.

The LCM of the given fractions is

\frac{72}{5}

.

The HCF of the given fractions is

\frac{72}{5}

.

The LCM of the given fractions is

\frac{6}{1925}

.

Cevabı ve açıklamayı göster

Cevap: The HCF of the given fractions is $\frac{6}{1925}$ .; The LCM of the given fractions is $\frac{72}{5}$ .

Cevap

The correct statements are that the HCF of the fractions is

\frac{6}{1925}

and the LCM of the fractions is

\frac{72}{5}

.

The correct statements properly evaluate the formulas for fractions: the HCF of a set of fractions is the HCF of their numerators divided by the LCM of their denominators, and the LCM of a set of fractions is the LCM of their numerators divided by the HCF of their denominators.

Adım Adım Çözüm

1

Find the HCF and LCM of the numerators (

12, 18, 24

).

HCF(

12, 18, 24

) =

6

, and LCM(

12, 18, 24

) =

72

.

These values are required for the numerator positions in the fraction HCF and LCM formulas.

2

Find the HCF and LCM of the denominators (

25, 35, 55

).

HCF(

25, 35, 55

) =

5

, and LCM(

25, 35, 55

) =

5^2 \times 7 \times 11 = 1925

.

These values are required for the denominator positions in the fraction HCF and LCM formulas.

3

Calculate the HCF of the fractions.

HCF =

\frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{6}{1925}

.

This determines the validity of the HCF statements.

4

Calculate the LCM of the fractions.

LCM =

\frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{72}{5}

.

This determines the validity of the LCM statements.

Anahtar Kavram

Highest Common Factor (HCF) and Least Common Multiple (LCM) of Fractions

Soru 51Soru →

An event organizer needs to pack $144$ apples, $180$ oranges, and $216$ bananas into identical fruit baskets. Each basket must contain only one type of fruit, and all baskets must contain exactly the same number of fruits. What is the minimum total number of fruit baskets the organizer must prepare?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the minimum number of baskets, we first need to determine the maximum number of fruits that can be placed in each basket. Since the number of fruits must be the same for all baskets and fruits cannot be mixed, this maximum number is the Highest Common Factor (HCF) of 144, 180, and 216. The HCF is 36. Dividing the total count of each fruit by 36 gives 4 apple baskets, 5 orange baskets, and 6 banana baskets. Adding these together yields a total of 15 baskets.

Adım Adım Çözüm

1

Determine the mathematical operation required to find the maximum number of fruits per basket.

The problem requires finding the Highest Common Factor (HCF) of 144, 180, and 216.

Because the baskets must be identical in capacity, hold only one type of fruit, and we want the minimum number of total baskets (meaning maximum fruits per basket).

2

Perform prime factorization for each fruit quantity.

144 = 2^4 * 3^2; 180 = 2^2 * 3^2 * 5; 216 = 2^3 * 3^3

Prime factorization is the most reliable method for finding the HCF of three large numbers.

3

Calculate the Highest Common Factor (HCF).

HCF = 2^2 * 3^2 = 4 * 9 = 36.

The HCF is the product of the lowest powers of common prime factors present in all three numbers.

4

Calculate the number of baskets required for each individual fruit type.

Apples: 144 / 36 = 4 baskets. Oranges: 180 / 36 = 5 baskets. Bananas: 216 / 36 = 6 baskets.

Dividing the total quantity of each fruit by the maximum capacity of one basket gives the basket count per fruit.

5

Calculate the total number of baskets.

4 + 5 + 6 = 15 baskets.

The question asks for the minimum total number of fruit baskets to be prepared.

Anahtar Kavram

Highest Common Factor (HCF) applied to equitable distribution and grouping

Soru 52Soru →

In a digital encryption protocol, a security key is generated based on a master integer $N$ . When $N$ is successively divided by $6$ , $8$ , and $9$ , the resulting remainders are $4$ , $3$ , and $7$ , respectively. If $N$ is the smallest possible positive integer that satisfies these conditions, what is the remainder when $N$ is divided by $19$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

By applying the rules of successive division and working backwards from a final quotient of 0, the smallest possible integer

N

is found to be 358. Dividing 358 by 19 yields a quotient of 18 and a remainder of 16.

Adım Adım Çözüm

1

Set up equations based on the definition of successive division.

N = 6q_1 + 4

,

q_1 = 8q_2 + 3

, and

q_2 = 9q_3 + 7

Successive division means each division is performed on the quotient of the previous step.

2

Determine the value of the final quotient

q_3

to minimize

N

.

q_3 = 0

The smallest possible positive initial number

N

is obtained when the final successive quotient is zero.

3

Solve for intermediate quotient

q_2

.

q_2 = 9(0) + 7 = 7

Substitute

q_3 = 0

into the equation

q_2 = 9q_3 + 7

.

4

Solve for intermediate quotient

q_1

.

q_1 = 8(7) + 3 = 59

Substitute

q_2 = 7

into the equation

q_1 = 8q_2 + 3

.

5

Calculate the smallest positive integer

N

.

N = 6(59) + 4 = 358

Substitute

q_1 = 59

into the first equation.

6

Divide

N

by

19

to find the final remainder.

358 = 19 \times 18 + 16

. The remainder is

16

.

The problem asks for the remainder when the resulting

N

is divided by

19

.

Anahtar Kavram

Successive Division and Remainder Theorem

Tahmini Süre:1m 30s

Soru 53Soru →

Three electronic metronomes are set to tick at regular intervals of $\frac{2}{3}$ of a second, $\frac{3}{4}$ of a second, and $\frac{4}{5}$ of a second, respectively. If they all tick simultaneously at a given moment, after how many seconds will they next tick together simultaneously?

12

\frac{1}{60}

\frac{1}{5}

1

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer accurately uses the formula for the LCM of fractions, which is the LCM of the numerators (

2, 3, 4

) divided by the HCF of the denominators (

3, 4, 5

). This results in

12 \div 1 = 12

seconds.

Adım Adım Çözüm

1

Identify the mathematical operation required.

To find when the metronomes tick together next, we must find the Least Common Multiple (LCM) of their interval times:

\frac{2}{3}

,

\frac{3}{4}

, and

\frac{4}{5}

.

The LCM of multiple time intervals gives the smallest total time at which all periodic events align.

2

Apply the formula for the LCM of fractions.

The formula is:

LCM = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

.

This standard formula ensures the resulting value is a multiple of all the given fractional intervals.

3

Calculate the LCM of the numerators.

The numerators are

2, 3,

and

4

. Their LCM is

12

.

12

is the smallest number perfectly divisible by

2

,

3

, and

4

.

4

Calculate the HCF of the denominators.

The denominators are

3, 4,

and

5

. Their Highest Common Factor (HCF) is

1

.

3, 4,

and

5

are co-prime integers with no common divisor other than

1

.

5

Determine the final LCM.

LCM = \frac{12}{1} = 12

seconds.

Substituting the calculated values into the formula yields the final answer.

Anahtar Kavram

LCM of Fractions

Tahmini Süre:45s

Soru 54Soru →

A data verification algorithm analyzes inputs based on their fundamental mathematical classifications. During a specific test run, the system is fed four distinct variables:

- $w = \frac{22}{7}$
- $x =$ an arbitrary irrational number
- $y =$ the additive identity of the real number system
- $z =$ an arbitrary prime number strictly greater than 2

Based on the strict mathematical definitions and properties of numbers, which of the following statements are ALWAYS true? (Select all that apply)

Geçerli olan tümünü seçin

The product

x \cdot w

is guaranteed to evaluate to an irrational number.

The sum

z + y

will always result in a positive odd integer.

The quotient

\frac{\pi}{w}

evaluates to exactly

1

, as both terms represent the same mathematical constant.

The variable

y

belongs to the set of positive integers.

Cevabı ve açıklamayı göster

Cevap: The product $x \cdot w$ is guaranteed to evaluate to an irrational number.; The sum $z + y$ will always result in a positive odd integer.

Cevap

The statements confirming that the product of

x

and

w

is irrational, and that the sum of

z

and

y

is a positive odd integer, are the mathematically correct conclusions.

The correct statements accurately apply the closure rule that a non-zero rational multiplied by an irrational is irrational, and correctly identify that any prime greater than 2 is a positive odd integer whose value is unchanged when adding zero.

Adım Adım Çözüm

1

Classify the variable

w

and evaluate its product with

x

.

The number

w = \frac{22}{7}

is a non-zero rational number. The product of a non-zero rational number and an irrational number (

x

) is always irrational.

This tests the closure properties of rational and irrational numbers under multiplication.

2

Identify the value of

y

and evaluate its classification.

The additive identity of the real number system is

0

. The number

0

is an even integer, but it is neither positive nor negative.

This ensures the exact boundary conditions of integer classifications (positive, negative, zero) are understood.

3

Determine the properties of

z

and evaluate the sum

z + y

.

All prime numbers strictly greater than

2

are odd and positive. Since

y = 0

, the sum

z + 0 = z

remains a positive odd integer.

This applies the parity rules of prime numbers.

4

Compare

\pi

and

w

to evaluate their quotient.

While

\frac{22}{7}

is a common approximation for

\pi

,

\pi

is irrational and

\frac{22}{7}

is rational. They are not equal, so their quotient is not

1

.

This distinguishes between exact mathematical constants and their real-world practical approximations.

Anahtar Kavram

Classification of Real Numbers, Closure Properties, Parity of Primes, and Properties of Zero

Soru 55Soru →

When the positive integer $N$ is divided by $15$ , $20$ , and $24$ , it leaves a remainder of $8$ in each case. If $N$ is the smallest such number that is also exactly divisible by $13$ , what is the value of $N$ ?

832

128

728

1664

Cevabı ve açıklamayı göster

Cevap: 728

Cevap

728

The Lowest Common Multiple (LCM) of 15, 20, and 24 is 120. Any number leaving a remainder of 8 when divided by these numbers can be expressed in the form 120k + 8. By testing integer values for k, we find that when k = 6, the number is 120(6) + 8 = 728. Since 728 is perfectly divisible by 13 (728 / 13 = 56), it satisfies all conditions and is the correct answer.

Adım Adım Çözüm

1

Find the Lowest Common Multiple (LCM) of the given divisors: 15, 20, and 24.

LCM(15, 20, 24) = 120.

The base number without the remainder must be exactly divisible by all three numbers.

2

Set up a general algebraic expression for the number

N

.

N = 120k + 8, where k is a positive integer.

Because the number leaves a remainder of 8 in each case, it must be 8 more than a multiple of their LCM.

3

Apply the final divisibility condition to find the smallest valid

k

.

Test values of k in the expression (120k + 8) to see which result is divisible by 13.

The problem specifies that the final number

N

must be exactly divisible by 13.

4

Evaluate

120k + 8

for

k = 1, 2, 3...

until a multiple of 13 is reached.

For k = 6, N = 120(6) + 8 = 728. Checking divisibility: 728 / 13 = 56 (an exact integer).

Sequential testing guarantees we find the smallest possible value for

N

.

Anahtar Kavram

Applications of LCM and Modular Arithmetic

Soru 56Soru →

Compute the exact Highest Common Factor (HCF) for the fractions $\frac{6}{25}$ and $\frac{9}{10}$ . Express your final result as a decimal.

Cevabı ve açıklamayı göster

Cevap: 0.06

Cevap

0.06

The Highest Common Factor (HCF) of a set of fractions is found by dividing the HCF of their numerators by the LCM of their denominators. In this case, HCF(6, 9) is 3, and LCM(25, 10) is 50. This gives the fraction 3/50, which perfectly evaluates to the decimal 0.06.

Adım Adım Çözüm

1

Identify the standard rule for calculating the HCF of fractional numbers.

HCF = (HCF of numerators) / (LCM of denominators)

This is the mathematical formula required to find the greatest common divisor of multiple fractions.

2

Find the Highest Common Factor (HCF) of the two numerators, 6 and 9.

HCF(6, 9) = 3

3 is the largest integer that divides both 6 and 9 without leaving a remainder.

3

Find the Least Common Multiple (LCM) of the two denominators, 25 and 10.

LCM(25, 10) = 50

50 is the smallest positive integer that is a multiple of both 25 and 10.

4

Substitute the results into the formula and convert the fraction to a decimal.

3 / 50 = 0.06

Dividing 3 by 50 yields the terminating decimal 0.06, which is the final required format.

Anahtar Kavram

HCF and LCM of fractions

Soru 57Soru →

Consider the integer $N = 4 - 2 \times 2^{68} + 7 \times 14^{85}$ . If $N$ is divided by $13$ , which of the following represents the correct remainder?

6

7

5

9

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The correct remainder is 6.

By applying Fermat's Little Theorem,

2^{68} \equiv 2^8 \equiv 9 \pmod{13}

, and clearly

14^{85} \equiv 1^{85} \equiv 1 \pmod{13}

. Substituting these values gives the expression

4 - 2(9) + 7(1) = 4 - 18 + 7 = -7

. To find the valid positive remainder, we add the divisor 13 to the negative result, yielding

-7 + 13 = 6

.

Adım Adım Çözüm

1

Simplify the term

14^{85} \pmod{13}

1

Since

14 \equiv 1 \pmod{13}

, any power of 14 will also be congruent to

1^{85} = 1

.

2

Simplify the term

2^{68} \pmod{13}

using Fermat's Little Theorem

9

Fermat's theorem states

a^{12} \equiv 1 \pmod{13}

. Dividing 68 by 12 gives a quotient of 5 and a remainder of 8. Thus,

2^{68} \equiv 2^8 \pmod{13}

. Since

2^8 = 256

, and

256 = 13 \times 19 + 9

, the value is 9.

3

Substitute the simplified modulo values back into the original expression

4 - 2(9) + 7(1) \pmod{13}

Replacing the large exponent terms with their modulo 13 equivalents allows for simple arithmetic evaluation.

4

Evaluate the simplified arithmetic expression from left to right

-7

Following the correct order of operations:

4 - 18 + 7 = -14 + 7 = -7

.

5

Convert the negative result into a valid positive remainder

6

Remainders must be non-negative. Add the divisor (13) to the negative result:

-7 + 13 = 6

.

Anahtar Kavram

Modular Arithmetic, Fermat's Little Theorem, and Negative Remainders

Soru 58Soru →

Consider a sequence of integers generated by the mathematical formula $T_n = 17^n - 40$ . What is the positive remainder obtained when the $123^{\text{rd}}$ term of this sequence is divided by $9$ ?

4

5

6

3

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Applying the rules of modular arithmetic, we first reduce the base

17

modulo

9

, giving

-1

. Raising

-1

to the odd power of

123

yields

-1

. The expression then simplifies to

-1 - 40 = -41

. Because remainders must be non-negative, we add the next largest multiple of

9

(which is

45

) to

-41

, resulting in a final positive remainder of

4

.

Adım Adım Çözüm

1

Set up the expression for the

123^{\text{rd}}

term modulo

9

.

T_{123} \equiv (17^{123} - 40) \pmod 9

We need to find the remainder of the specific sequence term when divided by

9

.

2

Reduce the base of the exponent modulo

9

.

17 \equiv -1 \pmod 9

Using a base of

-1

simplifies the calculation of large powers.

3

Evaluate the exponent part using the reduced base.

(-1)^{123} = -1

Raising a negative number to an odd power retains the negative sign.

4

Substitute back into the expression and compute the intermediate remainder.

-1 - 40 = -41

Combining the evaluated power with the rest of the sequence formula.

5

Convert the negative value into a valid positive remainder modulo

9

.

-41 = 9 \times (-5) + 4 \implies \text{Remainder is } 4

A true remainder must be a positive integer strictly less than the divisor. We achieve this by adding multiples of

9

until the value is positive.

Anahtar Kavram

Modular arithmetic reduction, exponent parity rules, and conversion of negative remainders.

Soru 59Soru →

Consider the following statements regarding the classification and properties of numbers:

I. The number $0$ is the smallest positive even integer, and it can be produced by calculating the sum of two distinct irrational numbers.
II. The rational fraction $\frac{22}{7}$ represents the exact value of the mathematical constant $\pi$ on the real number line.
III. Every prime number greater than $2$ is an odd integer, and the square of any odd integer always leaves a remainder of $1$ when divided by $8$ .

Which of the statements given above is/are correct?

Only III

Only I and III

Only II and III

I, II, and III

Cevabı ve açıklamayı göster

Cevap: Only III

Cevap

Only Statement III is correct. Statement I fails because 0 is not positive, and Statement II fails because pi is irrational and not exactly equal to 22/7.

The correct answer accurately isolates Statement III as the only mathematically true proposition. It correctly identifies the parity of prime numbers and the modulo 8 behavior of odd squares, while dismissing the flawed definitions of zero and

\pi

present in the other statements.

Adım Adım Çözüm

1

Evaluate Statement I regarding the properties of the number

0

and the sum of irrationals.

Statement I is false.

While the sum of two distinct irrational numbers (e.g.,

\sqrt{2}

and

-\sqrt{2}

) can indeed be

0

, the number

0

itself is neither positive nor negative. Describing it as the 'smallest positive even integer' is incorrect. The smallest positive even integer is

2

.

2

Evaluate Statement II regarding the relationship between

\pi

and

\frac{22}{7}

.

Statement II is false.

The constant

\pi

is an irrational number (a non-terminating, non-repeating decimal), while

\frac{22}{7}

is a rational number. They are not exactly equal;

\frac{22}{7}

is merely a convenient fractional approximation used in basic calculations.

3

Evaluate Statement III regarding primes and the square of odd integers.

Statement III is true.

The only even prime is

2

, so all primes greater than

2

are odd. For any odd integer

n = 2k + 1

, its square is

(2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1

. Since

k(k+1)

is the product of two consecutive integers, it must be even (let

k(k+1) = 2m

). Thus,

4(2m) + 1 = 8m + 1

, proving it always leaves a remainder of

1

when divided by

8

.

4

Determine the correct option based on the evaluations.

The option stating 'Only III' is the correct choice.

Since Statements I and II contain mathematical misconceptions, Statement III is the only accurate assertion.

Anahtar Kavram

Classification of numbers including rational versus irrational properties, integer parity, and fundamental prime characteristics.

Soru 60Soru →

Consider the positive integers $M = 360$ and $N = 504$ . Based on the mathematical properties of factors and multiples, which of the following statements are correct?

Geçerli olan tümünü seçin

The Least Common Multiple (LCM) of

M

and

N

is

2520

.

The Highest Common Factor (HCF) of

M

and

N

is

72

.

The integer

M

has exactly

24

distinct positive prime factors.

The Highest Common Factor (HCF) of the fractions

\frac{M}{7}

and

\frac{N}{5}

is

2520

.

Cevabı ve açıklamayı göster

Cevap: The Least Common Multiple (LCM) of $M$ and $N$ is $2520$ .; The Highest Common Factor (HCF) of $M$ and $N$ is $72$ .

Cevap

The statements confirming the Least Common Multiple (LCM) is 2520 and the Highest Common Factor (HCF) is 72 are mathematically correct.

Based on the prime factorizations

M = 2^3 \times 3^2 \times 5

and

N = 2^3 \times 3^2 \times 7

, the HCF is derived from the lowest common powers (

2^3 \times 3^2 = 72

), and the LCM is derived from the highest powers of all prime bases (

72 \times 5 \times 7 = 2520

). Thus, the statements asserting these two values are correct.

Adım Adım Çözüm

1

Determine the prime factorizations of M and N.

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

and

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

.

Prime factorization is the foundational step required to find factors, HCF, and LCM accurately.

2

Calculate the Highest Common Factor (HCF) and Least Common Multiple (LCM) of M and N.

HCF =

2^3 \times 3^2 = 72

. LCM =

2^3 \times 3^2 \times 5 \times 7 = 2520

.

This confirms the validity of the first two statements.

3

Evaluate the number of factors for integer M.

Total positive factors =

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

. However, distinct prime factors are only 2, 3, and 5 (total of 3).

This shows the statement claiming 24 prime factors is factually incorrect due to a specific modifier (prime).

4

Calculate the HCF of the given fractions.

HCF =

\frac{\text{HCF}(360, 504)}{\text{LCM}(7, 5)} = \frac{72}{35}

.

This invalidates the final statement, which improperly applied the LCM fraction formula.

Anahtar Kavram

Applying prime factorization to calculate HCF, LCM, total factor counts, and properties of fractions.

Basic Numeracy

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