Pre-Algebra Alıştırma Soruları — ACT — Sayfa 4

Soru 61Soru →

A positive integer $n$ has the prime factorization $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The greatest common divisor of $n$ and $840$ is $120$ , and the least common multiple of $n$ and $90$ is $1,800$ . If $n$ is not divisible by $9$ , what is the value of $a + b + c$ ?

5

6

7

8

9

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is 6. By analyzing the prime factorizations, the greatest common divisor condition

\text{GCD}(n, 840) = 120

tells us that the exponents of the prime factors of

n

satisfy

a \geq 3

,

b \geq 1

, and

c \geq 1

. The least common multiple condition

\text{LCM}(n, 90) = 1,800

tells us that

a = 3

,

b \leq 2

, and

c = 2

. Finally, the condition that

n

is not divisible by 9 requires

b < 2

, which forces

b = 1

. Summing these values gives

3 + 1 + 2 = 6

.

Adım Adım Çözüm

1

Find the prime factorizations of the given numbers.

840 = 2^3 \times 3^1 \times 5^1 \times 7^1

,

120 = 2^3 \times 3^1 \times 5^1

,

90 = 2^1 \times 3^2 \times 5^1

, and

1,800 = 2^3 \times 3^2 \times 5^2

.

Decomposing the numbers into their prime factors allows us to apply the rules of Greatest Common Divisor (GCD) and Least Common Multiple (LCM) on the exponents.

2

Analyze the GCD condition.

Since

\text{GCD}(n, 840) = 120

, we have

\min(a, 3) = 3

,

\min(b, 1) = 1

, and

\min(c, 1) = 1

. This implies

a \geq 3

,

b \geq 1

, and

c \geq 1

.

The GCD of two numbers takes the minimum exponent for each prime factor present in both numbers.

3

Analyze the LCM condition.

Since

\text{LCM}(n, 90) = 1,800

, we have

\max(a, 1) = 3

,

\max(b, 2) = 2

, and

\max(c, 1) = 2

. This implies

a = 3

,

b \leq 2

, and

c = 2

.

The LCM of two numbers takes the maximum exponent for each prime factor present in either number.

4

Apply the divisibility constraint to determine the exponents.

From previous steps,

a = 3

and

c = 2

. The exponent

b

must satisfy

1 \leq b \leq 2

. Since

n

is not divisible by

9 = 3^2

, the exponent of 3 in the factorization of

n

must be strictly less than 2. Thus,

b = 1

.

If

b = 2

, then

n

would contain

3^2

, making it divisible by 9, which violates the given constraint.

5

Calculate the sum

a + b + c

.

a + b + c = 3 + 1 + 2 = 6

.

Adding the determined exponents yields the final requested value.

Anahtar Kavram

Using prime factorizations to find the greatest common divisor and least common multiple of integers.

Alternatif Yöntem

Instead of checking each prime factor independently, we can find the value of

n

directly. Since

\text{GCD}(n, 840) = 120

,

n

must be a multiple of

120

. The multiples of

120

are

120, 240, 360, 480, 600, 720, 840, \dots

. We test which of these multiples satisfies

\text{LCM}(n, 90) = 1,800

and is not divisible by

9

. Since

1,800 = 15 \times 120

,

n

must divide

1,800

. The divisors of

1,800

that are multiples of

120

are

120, 240, 360, 600, 900, 1,800

. Among these, only

600

has a greatest common divisor of

120

with

840

, is not divisible by

9

, and has

\text{LCM}(600, 90) = 1,800

. Thus,

n = 600 = 2^3 \times 3^1 \times 5^2

, which gives

a+b+c = 3+1+2 = 6

.

Tahmini Süre:2m 0s

Soru 62Soru →

A certain type of single-celled organism has a length of approximately $2.5 \times 10^{-6}$ meters. If $4 \times 10^3$ of these organisms are placed end-to-end in a straight line, what is the total length of the line, in meters, written in scientific notation?

1.0 \times 10^{-4}

1.0 \times 10^{-3}

1.0 \times 10^{-2}

1.0 \times 10^{-8}

1.0 \times 10^{-17}

Cevabı ve açıklamayı göster

Cevap: $1.0 \times 10^{-2}$

Cevap

The total length of the line is

1.0 \times 10^{-2}

meters.

To find the total length of the line of organisms, multiply the length of a single organism by the total number of organisms:

(2.5 \times 10^{-6}) \times (4 \times 10^3)

. Grouping the coefficients and powers of ten gives

(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-3}

. To write this in standard scientific notation, rewrite

10

as

1.0 \times 10^1

, which yields

1.0 \times 10^{1 + (-3)} = 1.0 \times 10^{-2}

.

Adım Adım Çözüm

1

Set up the multiplication of the two values to find the total length.

Total Length

= (2.5 \times 10^{-6}) \times (4 \times 10^3)

To find the combined length of multiple organisms placed end-to-end, multiply the length of one organism by the total number of organisms.

2

Group the coefficients and the powers of

10

together, then perform the multiplication.

(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-6 + 3} = 10 \times 10^{-3}

Use the associative and commutative properties of multiplication, and add the exponents when multiplying powers with the same base:

10^a \times 10^b = 10^{a+b}

.

3

Convert the result into standard scientific notation

a \times 10^n

where

1 \le |a| < 10

.

10 \times 10^{-3} = 1.0 \times 10^1 \times 10^{-3} = 1.0 \times 10^{-2}

Since

10

is not less than

10

, rewrite

10

as

1.0 \times 10^1

and add the exponents (

1 + (-3) = -2

) to express the number in standard scientific notation.

Anahtar Kavram

Multiplying numbers in scientific notation and converting to standard scientific notation.

Tahmini Süre:1m 0s

Soru 63Soru →

A basket contains red apples and green apples in a ratio of $2:5$ . If there are 35 apples in the basket, how many green apples are in the basket?

5

10

14

25

7

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

To find the number of green apples, first determine the total number of parts in the ratio by adding the two components:

2 + 5 = 7

parts. Next, divide the total number of apples in the basket by the total parts to find the value of each part:

35 \div 7 = 5

apples per part. Finally, multiply the number of parts representing green apples (

5

parts) by the value of each part (

5

):

5 \times 5 = 25

. Therefore, the correct quantity of green apples in the basket is 25.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

2 + 5 = 7

parts

The ratio of red apples to green apples is

2:5

, so the total quantity is divided into

7

equal parts.

2

Calculate the value of a single part of the ratio.

35 \div 7 = 5

apples per part

Divide the total number of apples (

35

) by the total number of parts (

7

) to find how many apples make up one part.

3

Multiply the value of a single part by the number of parts representing green apples.

5 \text{ parts} \times 5 \text{ apples/part} = 25

green apples

Since green apples represent

5

parts of the ratio, multiply

5

by the value of one part (

5

) to find the total number of green apples.

Anahtar Kavram

Solving word problems involving part-to-part ratios and totals by finding the unit value per ratio part.

Soru 64Soru →

A positive integer $K$ has exactly $12$ positive factors. The greatest common divisor of $K$ and $84$ is $6$ , and the least common multiple of $K$ and $24$ is $360$ . What is the value of $K$ ?

60

72

90

120

180

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90

The prime factorizations are

84 = 2^2 \times 3 \times 7

,

6 = 2 \times 3

,

24 = 2^3 \times 3

, and

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

. The condition

\gcd(K, 84) = 6

implies the power of 2 in

K

is exactly 1, and the power of 3 is at least 1. The condition

\text{lcm}(K, 24) = 360

implies the power of 3 in

K

is exactly 2, and the power of 5 is exactly 1. Thus,

K

must be

2^1 \times 3^2 \times 5^1 = 90

. Testing

90

, it has

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

positive factors, which perfectly satisfies all conditions.

Adım Adım Çözüm

1

Find the prime factorization of the given numbers in the problem.

84 = 2^2 \times 3 \times 7

,

6 = 2 \times 3

,

24 = 2^3 \times 3

, and

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

.

Expressing these numbers in terms of their prime components allows us to determine the constraints on the prime factorization of

K

.

2

Analyze the greatest common divisor constraint

\gcd(K, 84) = 6

.

The prime 2 must have an exponent of exactly 1 in the factorization of

K

(since

84

has

2^2

and the greatest common divisor has only

2^1

). The prime 3 must have an exponent of at least 1 in

K

. The prime 7 cannot be a factor of

K

.

The greatest common divisor selects the minimum exponent for each shared prime factor.

3

Analyze the least common multiple constraint

\text{lcm}(K, 24) = 360

.

The prime 3 must have an exponent of exactly 2 in

K

(since

24

has

3^1

and the least common multiple has

3^2

). The prime 5 must have an exponent of exactly 1 in

K

(since

24

has

5^0

and the least common multiple has

5^1

).

K

cannot contain any prime factors other than 2, 3, and 5.

The least common multiple selects the maximum exponent for each prime factor present in either number.

4

Combine the exponent constraints to determine

K

and verify its factor count.

K = 2^1 \times 3^2 \times 5^1 = 90

. The number of positive factors of

90

is

(1+1)(2+1)(1+1) = 2 \times 3 \times 2 = 12

.

This unique configuration matches all given greatest common divisor and least common multiple prime power requirements and satisfies the factor count condition.

Anahtar Kavram

Prime Factorization, Greatest Common Divisor, Least Common Multiple, and Number of Factors

Alternatif Yöntem

Instead of analyzing prime factor exponents, check the given multiple-choice options. Test each option by checking if it has exactly 12 factors, a greatest common divisor of 6 with 84, and a least common multiple of 360 with 24. Only the value 90 meets all three criteria.

Tahmini Süre:2m 30s

Soru 65Soru →

A party planner is preparing gift bags. She has a box containing 12 toy cars and another box containing 18 stickers. She wants to purchase the minimum number of boxes of each toy so that she has the exact same number of toy cars and stickers, with no items left over. If she puts all of these purchased items into the gift bags, what is the minimum total number of items (cars and stickers combined) she will have?

6

12

30

36

72

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72

To find the minimum total number of items, we must first find the least common multiple (LCM) of the box sizes, 12 and 18. The prime factorizations are

12 = 2^2 \times 3

and

18 = 2 \times 3^2

. The LCM is

2^2 \times 3^2 = 4 \times 9 = 36

. This means the planner must buy 36 toy cars (3 boxes of 12) and 36 stickers (2 boxes of 18) to have an equal amount of each. The minimum total number of items combined is

36 + 36 = 72

.

Adım Adım Çözüm

1

Identify that the number of toy cars and stickers must be equal and must be a multiple of the box sizes (12 and 18). Find the least common multiple (LCM) of 12 and 18 to determine the minimum number of each item needed.

LCM(12, 18) = 36

Since the planner wants the minimum number of boxes and equal amounts of both items, the number of each item must be the smallest common multiple of the two box sizes.

2

Calculate the total number of items by adding the number of toy cars and stickers together.

36 + 36 = 72

The question asks for the total number of items combined, so we must add the 36 toy cars to the 36 stickers.

Anahtar Kavram

Least Common Multiple (LCM)

Tahmini Süre:1m 30s

Soru 66Soru →

Evaluate the four mathematical expressions below, and arrange them in order from least to greatest value.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order of expressions from least to greatest is the expression evaluating to

-25

, followed by the expression evaluating to

-8

, then the expression evaluating to

-4

, and finally the expression evaluating to

16

.

Evaluating each expression using the correct order of operations yields

-25

for the expression containing the negative base exponent,

-8

for the expression containing the absolute value,

-4

for the expression containing the cubed term, and

16

for the expression containing the squared term. Ordering these values from least to greatest results in the sequence

-25 < -8 < -4 < 16

.

Adım Adım Çözüm

1

Evaluate the first expression:

2 \times (-3)^2 - 4^2 \div 8

.

Value is

16

.

Exponents are calculated first:

(-3)^2 = 9

and

4^2 = 16

. Then, perform multiplication and division:

2 \times 9 = 18

and

16 \div 8 = 2

. Finally, subtract:

18 - 2 = 16

.

2

Evaluate the second expression:

-2^4 + 3 \times (5 - 8)

.

Value is

-25

.

Parentheses are evaluated first:

5 - 8 = -3

. Next, evaluate the exponent:

-2^4 = -16

(since the exponent applies to the base

2

before negation). Perform multiplication:

3 \times (-3) = -9

. Finally, add:

-16 + (-9) = -25

.

3

Evaluate the third expression:

(3 - 5)^3 \div 4 - 2

.

Value is

-4

.

Parentheses are evaluated first:

3 - 5 = -2

. Next, cube the result:

(-2)^3 = -8

. Perform division:

-8 \div 4 = -2

. Finally, subtract:

-2 - 2 = -4

.

4

Evaluate the fourth expression:

-| -6 - 2^2 \times 3 | + 10

.

Value is

-8

.

Within the absolute value bars, evaluate the exponent first:

2^2 = 4

, then multiply:

4 \times 3 = 12

. Subtract to find the absolute value interior:

-6 - 12 = -18

. The absolute value of

-18

is

18

, which is negated to

-18

. Finally, add

10

to get

-18 + 10 = -8

.

5

Order the final values from least to greatest.

-25 < -8 < -4 < 16

.

Comparing the values

-25

,

-8

,

-4

, and

16

on the number line determines their relative size.

Anahtar Kavram

Order of operations rules dictate that expressions are evaluated in the sequence of Parentheses/Grouping symbols, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Unary negation is performed after exponentiation unless grouped by parentheses.

Tahmini Süre:1m 30s

Soru 67Soru →

An expression is shown below.

\frac{2^3 \times 2^2 - 4 \times (1 - 3)^3}{|2 - 2^3| - 10}

What is the value of this expression when it is simplified using the standard order of operations?

-24

-16

-4

0

56

Cevabı ve açıklamayı göster

Cevap: -16

Cevap

The correct answer is

-16

.

To find the value of the expression, follow the standard order of operations (PEMDAS). First, simplify terms with exponents and parentheses:

2^3 = 8

,

2^2 = 4

, and

(1 - 3)^3 = (-2)^3 = -8

. The numerator becomes

8 \times 4 - 4 \times (-8) = 32 - (-32) = 64

. In the denominator, evaluate the exponent first to get

2 - 8 = -6

, then apply the absolute value to get

|-6| = 6

, and finally subtract

10

to get

6 - 10 = -4

. Dividing the numerator by the denominator gives

\frac{64}{-4} = -16

.

Adım Adım Çözüm

1

Evaluate the exponents in the numerator and the denominator, including the term inside the parenthesis.

In the numerator,

2^3 = 8

,

2^2 = 4

, and

(1 - 3)^3 = (-2)^3 = -8

. In the denominator,

2^3 = 8

.

According to the order of operations (PEMDAS), operations inside grouping symbols (parentheses) and exponentiation must be performed before multiplication, division, addition, or subtraction.

2

Perform the multiplications in the numerator and evaluate the expression inside the absolute value in the denominator.

The numerator becomes

8 \times 4 - 4 \times (-8) = 32 - (-32)

. The denominator becomes

|2 - 8| - 10 = |-6| - 10

.

Multiplication has priority over addition and subtraction. In the denominator, the expression inside the absolute value acts as a grouping symbol and is simplified first.

3

Perform addition and subtraction in both the numerator and the denominator, including applying the absolute value.

The numerator is

32 + 32 = 64

. The denominator is

6 - 10 = -4

.

Subtracting a negative number is equivalent to adding its positive counterpart:

32 - (-32) = 32 + 32

. The absolute value of

-6

is

6

, so the denominator simplifies to

6 - 10 = -4

.

4

Divide the simplified numerator by the simplified denominator.

\frac{64}{-4} = -16

.

The fraction bar represents division, which is the final operation to perform to find the value of the expression.

Anahtar Kavram

Order of Operations and Number Properties

Alternatif Yöntem

Instead of evaluating

2^3 \times 2^2

separately, the laws of exponents can be used to simplify the term first:

2^3 \times 2^2 = 2^{3+2} = 2^5 = 32

.

Tahmini Süre:1m 30s

Soru 68Soru →

What is the value of the expression below?

-[(-3)^2 - |4 - 2 \times 5|] \times 2^3 - \frac{18 \div 3 \times (-2)}{-3 + 1}

Cevabı ve açıklamayı göster

Cevap: -30

Cevap

The value of the expression is

-30

.

Applying the correct order of operations (PEMDAS) ensures the expression is evaluated systematically. Evaluating the first term gives

-[(-3)^2 - |4 - 2 \times 5|] \times 2^3 = -[9 - |-6|] \times 8 = -[9 - 6] \times 8 = -3 \times 8 = -24

. Evaluating the fraction term gives

\frac{18 \div 3 \times (-2)}{-3 + 1} = \frac{6 \times (-2)}{-2} = \frac{-12}{-2} = 6

. Subtracting the fraction value from the first term results in

-24 - 6 = -30

.

Adım Adım Çözüm

1

Simplify the grouping symbols: the bracket and absolute value term.

-3

First, evaluate the exponent

(-3)^2 = 9

. Inside the absolute value, multiplication takes precedence over subtraction:

4 - 2 \times 5 = 4 - 10 = -6

. The absolute value is

|-6| = 6

. Subtracting this from

9

inside the brackets gives

3

. Finally, apply the negative sign in front of the brackets to get

-3

.

2

Evaluate the exponent

2^3

and perform the multiplication.

-24

Exponents are evaluated before multiplication. Since

2^3 = 8

, the multiplication becomes

-3 \times 8 = -24

.

3

Evaluate the fraction term.

6

In the numerator, multiplication and division are performed from left to right:

18 \div 3 = 6

, then

6 \times (-2) = -12

. The denominator evaluates to

-3 + 1 = -2

. Dividing the numerator by the denominator gives

\frac{-12}{-2} = 6

.

4

Perform the final subtraction.

-30

Subtract the evaluated fraction value from the first term:

-24 - 6 = -30

.

Anahtar Kavram

Order of operations (PEMDAS), absolute value, and exponent rules with signed numbers.

Alternatif Yöntem

Evaluate each major term separately, paying close attention to grouping symbols, negative signs, and the left-to-right order for multiplication and division.

Tahmini Süre:2m 30s

Soru 69Soru →

What is the correct order of the following numbers from least to greatest?

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least to greatest is:

6.8 \times 10^{-5}

,

7.1 \times 10^{-4}

,

1.5 \times 10^{-3}

, and

2.4 \times 10^{-3}

.

The correct order is determined by first arranging the values by their power of 10 exponent from least to greatest:

-5 < -4 < -3

. This places

6.8 \times 10^{-5}

as the smallest and

7.1 \times 10^{-4}

as the second smallest. For the remaining two values that share the exponent

-3

, we compare their coefficients:

1.5 < 2.4

, placing

1.5 \times 10^{-3}

before

2.4 \times 10^{-3}

.

Adım Adım Çözüm

1

Compare the exponents of the base 10 to establish the primary order of magnitude.

The exponents are

-5

,

-4

, and

-3

. Since

-5 < -4 < -3

, we know that any number with an exponent of

-5

is smaller than one with

-4

, which in turn is smaller than one with

-3

.

In scientific notation, a smaller exponent on the power of 10 indicates a smaller overall value because it represents shifting the decimal point further to the left.

2

Place the numbers with unique exponents in order.

The smallest number is

6.8 \times 10^{-5}

, followed by

7.1 \times 10^{-4}

.

These have the smallest exponents (

-5

and

-4

, respectively).

3

Compare the coefficients of the remaining numbers that share the same exponent.

Both

1.5 \times 10^{-3}

and

2.4 \times 10^{-3}

share the exponent

-3

. Comparing their coefficients,

1.5 < 2.4

, so

1.5 \times 10^{-3} < 2.4 \times 10^{-3}

.

When the powers of 10 are identical, the values can be compared directly by their coefficients.

Anahtar Kavram

Comparing and ordering numbers written in scientific notation by analyzing their powers of 10 and coefficients.

Soru 70Soru →

Let $x$ , $y$ , and $z$ be integers such that $|x| < |y| < |z|$ , $x + y + z = -3$ , and $xyz > 0$ . What is the smallest possible value of $|x - y| + |y - z| + |z - x|$ ?

6

8

10

12

14

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer is 12. To satisfy the given conditions, the product

xyz > 0

means either all three numbers are positive, or exactly two are negative. Since they sum to

-3

, they cannot all be positive. Letting two be negative and one positive, we find that the integer set

\{-1, 2, -4\}

satisfies

|-1| < |2| < |-4|

(or

1 < 2 < 4

), has a sum of

-3

, and a product of

8 > 0

. The sum of the absolute differences is

|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12

, which is the minimum possible value.

Adım Adım Çözüm

1

Analyze the sign constraints from the product condition

xyz > 0

.

Since the product of

x

,

y

, and

z

is positive, either all three integers are positive, or exactly two are negative and one is positive.

The product of three numbers is positive if there are zero or two negative factors.

2

Evaluate the case where all three integers are positive.

If all three are positive, then

x, y, z \geq 1

. This implies their sum

x+y+z \geq 3

, which contradicts the constraint

x+y+z = -3

. Thus, this case is impossible.

Positive integers cannot sum to a negative number.

3

Analyze the case where two integers are negative and one is positive, and express them in terms of their absolute values.

Let the negative integers be

-u

and

-v

(with

u > v > 0

) and the positive integer be

p > 0

. The sum constraint becomes

-u - v + p = -3

, which simplifies to

p = u + v - 3

. The absolute values are

u

,

v

, and

p

, which must be three distinct positive integers satisfying

|x| < |y| < |z|

.

This translates the constraints into positive integer variables representing the absolute values.

4

Determine the values of

u, v, p

that minimize the maximum difference between any two of the three integers

x, y, z

.

For

v = 1

: if

u = 4

, then

p = 2

. The absolute values

\{1, 2, 4\}

are distinct. The corresponding integers are

x = -1

,

y = 2

, and

z = -4

. Their sum is

-3

and product is

8 > 0

. The difference between the maximum and minimum values is

2 - (-4) = 6

. For

v = 2

: if

u = 4

, then

p = 3

. The absolute values

\{2, 3, 4\}

are distinct, giving the integers

-4, -2, 3

, with a maximum difference of

3 - (-4) = 7

. For higher values, the difference only increases.

The expression

|x - y| + |y - z| + |z - x|

is equal to twice the difference between the maximum and minimum of the three integers.

5

Calculate the minimum value of the expression.

Using the optimal set

x = -1, y = 2, z = -4

, the value of the expression is

|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12

.

This is the smallest possible sum of the absolute differences.

Anahtar Kavram

Using absolute values and sign analysis to solve system of constraints on integers and distances on a number line.

Soru 71Soru →

A positive integer is said to have exactly $8$ positive factors. What is the fifth smallest positive integer that satisfies this condition?

Cevabı ve açıklamayı göster

Cevap: 54

Cevap

The fifth smallest positive integer with exactly 8 positive factors is 54.

The fifth smallest positive integer with exactly 8 positive factors is 54. The number of factors of

n = p_1^{a_1} \cdots p_k^{a_k}

is

(a_1+1)\cdots(a_k+1) = 8

. The possible factorization shapes are

p^7

(smallest is 128),

p^3 q

(smallest are 24, 40, 54, 56, 88), and

p q r

(smallest are 30, 42, 66, 70). Sorting these gives 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, where 54 is the fifth value.

Adım Adım Çözüm

1

Relate the number of positive factors to the prime factorization of a positive integer.

The number of positive factors is

(a_1 + 1)(a_2 + 1) \cdots (a_k + 1) = 8

.

This formula counts all possible combinations of prime factors that form divisors.

2

Determine the possible prime factorization structures that yield exactly 8 factors.

The possible prime factor structures are

p^7

,

p^3 q

, and

p q r

for distinct primes

p

,

q

, and

r

.

These correspond to the integer factorizations of 8: 8,

4 \times 2

, and

2 \times 2 \times 2

.

3

List the smallest candidate values for each structure.

Candidates include 128 (for

p^7

); 24, 40, 54, 56, 88 (for

p^3 q

); and 30, 42, 66, 70 (for

p q r

).

Evaluating structures with the smallest available prime numbers (2, 3, 5, 7, etc.) yields the smallest positive integers.

4

Sort all the candidates in ascending order.

The sorted list of smallest values is 24, 30, 40, 42, 54, 56, 66, 70, 78, 88.

Sorting allows us to identify the fifth smallest value precisely.

5

Identify the fifth value in the sorted list.

The fifth element is 54.

Counting from the smallest value (24) to the fifth position yields 54.

Anahtar Kavram

Determining the number of factors of an integer from its prime factorization.

Soru 72Soru →

What integer represents the midpoint between $-9$ and $7$ on a standard number line?

8

1

-1

-2

16

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

The midpoint coordinate is

-1

.

The midpoint of two numbers on a number line is their mathematical average. Adding the coordinates

-9

and

7

gives a sum of

-2

. Dividing this sum by

2

yields the midpoint coordinate of

-1

.

Adım Adım Çözüm

1

Identify the coordinates of the two endpoints on the number line.

The endpoints are

-9

and

7

.

Before calculating the midpoint, we must identify the values of the two points given in the problem.

2

Find the average of the two coordinates using the midpoint formula

\frac{a + b}{2}

.

\frac{-9 + 7}{2} = \frac{-2}{2} = -1

.

The midpoint of any two coordinates on a number line is their arithmetic mean.

Anahtar Kavram

Finding the midpoint between two integers on a number line

Tahmini Süre:45s

Soru 73Soru →

What is the value of the mathematical expression below?

-18 \div 3 \times 2 - (6 - 8)^3

-60

-20

-6

-4

5

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

-4

Evaluating the expression according to the order of operations (PEMDAS) yields the correct value. First, simplify the expression within the parentheses to get

6 - 8 = -2

. Second, evaluate the exponent to get

(-2)^3 = -8

. Next, perform multiplication and division from left to right: first divide

-18

by

3

to get

-6

, then multiply by

2

to get

-12

. Finally, perform the subtraction:

-12 - (-8) = -12 + 8 = -4

.

Adım Adım Çözüm

1

Evaluate the expression inside the parentheses.

6 - 8 = -2

According to the order of operations (PEMDAS), operations inside parentheses must be evaluated first.

2

Evaluate the exponent.

(-2)^3 = -8

Exponents are evaluated next. A negative number raised to an odd power yields a negative result.

3

Perform the division.

-18 \div 3 = -6

Multiplication and division have equal priority and must be performed from left to right.

4

Perform the multiplication.

-6 \times 2 = -12

Continuing from left to right, perform the multiplication next.

5

Perform the subtraction.

-12 - (-8) = -12 + 8 = -4

Finally, perform the subtraction. Subtracting a negative number is equivalent to adding its positive counterpart.

Anahtar Kavram

Order of Operations (PEMDAS) with negative numbers and exponents

Tahmini Süre:1m 0s

Soru 74Soru →

On a map of a city, the scale is $0.5\text{ inches} = 12\text{ miles}$ . If the actual distance between two towns is $96\text{ miles}$ , what is the distance between the two towns on the map, in inches?

2

4

8

16

48

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4 inches

To find the distance on the map, we set up the proportion comparing map inches to actual miles:

\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}}

. Cross-multiplying gives

12x = 48

, which simplifies to

x = 4

. Therefore, the distance on the map is

4\text{ inches}

.

Adım Adım Çözüm

1

Set up a proportion to relate the map distance to the actual distance.

\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}}

where

x

represents the distance between the two towns on the map.

This establishes a direct proportion based on the map's scale.

2

Solve for the unknown variable

x

by cross-multiplying.

12 \cdot x = 0.5 \cdot 96

12x = 48

Cross-multiplication is the standard algebraic method to solve proportions.

3

Divide both sides of the equation by 12 to find

x

.

x = \frac{48}{12} = 4

Isolating the variable gives the final map distance.

Anahtar Kavram

Solving ratio scale problems by setting up and solving proportions.

Soru 75Soru →

Let $x$ and $y$ be integers such that $|x| \leq 4$ and $|y| \leq 4$ . How many distinct pairs of integers $(x, y)$ satisfy the inequality $||x| - |y|| \geq 2$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

There are 36 distinct pairs of integers

(x, y)

that satisfy the inequality.

By analyzing the possible values for

|x|

and

|y|

within the set

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

, we identify the 12 pairs that satisfy the inequality

||x| - |y|| \geq 2

. When mapping these absolute values back to the actual integer coordinates, we account for the single option when a coordinate is 0 and the two positive/negative options when a coordinate is non-zero. Summing these possibilities yields exactly 36 distinct pairs.

Adım Adım Çözüm

1

Determine the range of absolute values for the integers.

The possible values for

|x|

and

|y|

are

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

.

Since

x

and

y

are integers satisfying

-4 \leq x, y \leq 4

, their absolute values must be non-negative integers up to 4.

2

Find the pairs of absolute values

(|x|, |y|)

that satisfy the inequality.

The valid pairs are:

(0, 2), (0, 3), (0, 4), (1, 3), (1, 4), (2, 0), (2, 4), (3, 0), (3, 1), (4, 0), (4, 1), (4, 2)

.

We test all combinations of

u, v \in \{0, 1, 2, 3, 4\}

such that

|u - v| \geq 2

.

3

Calculate the number of integer coordinate pairs

(x, y)

corresponding to each absolute value pair.

Pairs with one zero value yield 2 integer solutions (e.g.,

u=0, v=2 \Rightarrow (0, 2), (0, -2)

). Pairs with both non-zero values yield 4 integer solutions (e.g.,

u=1, v=3 \Rightarrow (1, 3), (1, -3), (-1, 3), (-1, -3)

).

An absolute value of 0 corresponds to only 1 integer (

0

), whereas any positive absolute value

k

corresponds to 2 integers (

k

and

-k

).

4

Sum the number of integer pairs for all valid cases.

Total pairs =

(3 \times 2) + (3 \times 2) + (6 \times 4) = 6 + 6 + 24 = 36

.

There are 3 pairs with

u=0

(

6

solutions), 3 pairs with

v=0

(

6

solutions), and 6 pairs with both

u, v > 0

(

24

solutions).

Anahtar Kavram

Solving nested absolute value inequalities with integer constraints and counting solution pairs systematically.

Soru 76Soru →

A high-speed fiber-optic cable transmits data at a rate of $8.0 \times 10^7$ bytes per second. A digital archive containing a total of $3.2 \times 10^{11}$ bytes needs to be transmitted. How many seconds will it take to complete the transmission?

2.5 \times 10^{-4}

4.0 \times 10^4

4.0 \times 10^3

4.0 \times 10^{17}

2.56 \times 10^{19}

Cevabı ve açıklamayı göster

Cevap: $4.0 \times 10^3$

Cevap

The transmission will take

4.0 \times 10^3

seconds.

The correct answer represents the transmission time of

4.0 \times 10^3

seconds. This is determined by dividing the total data volume,

3.2 \times 10^{11}

bytes, by the rate,

8.0 \times 10^7

bytes per second. The division yields

\frac{3.2}{8.0} \times 10^{11 - 7} = 0.4 \times 10^4

. Converting this value to standard scientific notation by moving the decimal point one digit to the right reduces the power of

10

by one, resulting in

4.0 \times 10^3

.

Adım Adım Çözüm

1

Set up the division to find the transmission time.

\text{Time} = \frac{3.2 \times 10^{11}\text{ bytes}}{8.0 \times 10^7\text{ bytes/second}}

To find the total time, divide the total quantity of data by the transmission rate.

2

Divide the numerical coefficients.

\frac{3.2}{8.0} = 0.4

When dividing numbers written in scientific notation, divide the coefficients first.

3

Apply the quotient rule of exponents to the powers of 10.

\frac{10^{11}}{10^7} = 10^{11 - 7} = 10^4

Subtract the exponent in the denominator from the exponent in the numerator.

4

Combine the intermediate parts and convert to standard scientific notation.

0.4 \times 10^4 = 4.0 \times 10^3

Standard scientific notation requires the coefficient to be at least 1 but strictly less than 10. Shifting the decimal point one place to the right decreases the exponent by 1.

Anahtar Kavram

Dividing numbers in scientific notation and converting to standard form

Tahmini Süre:1m 0s

Soru 77Soru →

A cleaning mixture is prepared by combining bleach and water in a ratio of $1:8$ . If a container holds a total of $36$ cups of this mixture, how many cups of bleach were used?

4.5

8

4

32

28

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

4 cups

The correct answer is 4. The ratio of bleach to water is

1:8

, which means the entire mixture is divided into

1 + 8 = 9

equal parts. Bleach constitutes

\frac{1}{9}

of the overall mixture. By taking

\frac{1}{9}

of the

36

-cup total volume, we find that the amount of bleach used is

4

cups.

Adım Adım Çözüm

1

Determine the total parts in the mixture's ratio.

1 + 8 = 9

total parts

Since the ratio of bleach to water is

1:8

, the mixture is composed of

1

part bleach for every

8

parts water.

2

Express the bleach portion as a fraction of the total mixture.

\frac{1}{9}

The bleach represents

1

part out of the

9

total parts in the mixture.

3

Find the amount of bleach in the

36

-cup container.

\frac{1}{9} \times 36 = 4

cups

Multiply the fraction of bleach by the total volume of the cleaning mixture.

Anahtar Kavram

Determining a part from a part-to-part ratio and a total quantity

Soru 78Soru →

An artist has a rectangular sheet of stained glass that measures $84$ inches by $120$ inches. The artist wants to cut the sheet into congruent square tiles of the largest possible side length, such that there is no glass wasted. What is the total number of square tiles the artist will obtain?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The total number of square tiles the artist will obtain is 70.

To find the maximum side length of the square tiles without wasting any glass, we must find the greatest common factor (GCF) of the rectangular dimensions,

84

and

120

. The prime factorizations are

84 = 2^2 \times 3 \times 7

and

120 = 2^3 \times 3 \times 5

, which gives a GCF of

2^2 \times 3 = 12

inches. The number of square tiles that fit along the length is

84 \div 12 = 7

and along the width is

120 \div 12 = 10

. Multiplying these dimensions gives a total of

7 \times 10 = 70

square tiles.

Adım Adım Çözüm

1

Determine the prime factorizations of the dimensions of the sheet.

84 = 2^2 \times 3 \times 7

and

120 = 2^3 \times 3 \times 5

This allows finding the greatest common factor of the two side lengths.

2

Calculate the greatest common factor (GCF) of

84

and

120

.

\text{GCF}(84, 120) = 2^2 \times 3 = 12

The largest congruent squares that can tile the sheet without waste must have a side length equal to this GCF.

3

Divide each dimension of the sheet by the tile side length to find the number of tiles along each side.

84 \div 12 = 7

tiles along the width, and

120 \div 12 = 10

tiles along the length.

This determines the grid dimensions of the tiles.

4

Multiply the number of tiles along the width by the number of tiles along the length.

7 \times 10 = 70

The total number of square tiles is the product of the number of tiles along each dimension.

Anahtar Kavram

Greatest Common Factor (GCF) application to division of a two-dimensional grid

Soru 79Soru →

Two positive integers, $a$ and $b$ , are such that $a < b$ . The greatest common divisor of $a$ and $b$ is $12$ , and their least common multiple is $720$ . If $a$ is a multiple of $5$ but $b$ is not a multiple of $5$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

By representing

a = 12x

and

b = 12y

with

\gcd(x, y) = 1

and

x < y

, the least common multiple constraint gives

12xy = 720

, which simplifies to

xy = 60

. Since

a

is a multiple of

5

and

b

is not, the factor of

5

in

60

must belong to

x

. Given that

x < y

and

\gcd(x, y) = 1

, the only valid coprime factorization of

60

where the factor

5

is in

x

and

x < y

is

x = 5

and

y = 12

. This yields

a = 12 \times 5 = 60

.

Adım Adım Çözüm

1

Express the two numbers in terms of their greatest common divisor (GCD).

a = 12x

and

b = 12y

, where

\gcd(x, y) = 1

and

x < y

.

Since the greatest common divisor of

a

and

b

is

12

, both numbers must be multiples of

12

, and their remaining parts

x

and

y

must be coprime to ensure their GCD is exactly

12

.

2

Use the least common multiple (LCM) to find the product of

x

and

y

.

xy = 60

.

The least common multiple of

12x

and

12y

when

\gcd(x, y) = 1

is

12xy

. Setting

12xy = 720

and dividing by

12

gives

xy = 60

.

3

Identify the constraints on

x

and

y

based on the divisibility by

5

.

5

must divide

x

, and

5

must not divide

y

.

We are given that

a = 12x

is a multiple of

5

, which means

5

must be a factor of

x

. Since

b = 12y

is not a multiple of

5

,

5

cannot be a factor of

y

.

4

Determine the unique pair

(x, y)

that satisfies all constraints.

x = 5

and

y = 12

.

The product

xy = 60

has prime factorization

2^2 \times 3 \times 5

. Since

\gcd(x, y) = 1

, the factor

5

must belong to

x

, and the other prime factors

2^2

and

3

can be distributed. To satisfy

x < y

, the only possible assignment is

x = 5

and

y = 12

(since other assignments like

x = 15, y = 4

or

x = 20, y = 3

violate

x < y

).

5

Calculate the value of

a

.

a = 60

.

Since

a = 12x

and

x = 5

, we find

a = 12 \times 5 = 60

.

Anahtar Kavram

Using prime factorizations to analyze greatest common divisors and least common multiples under algebraic and inequality constraints.

Soru 80Soru →

Point $P$ is located at $-14$ on a standard number line. Point $Q$ is located $8$ units from point $P$ in the positive direction. Point $R$ is located $11$ units from point $Q$ in the negative direction. What integer represents the location of point $R$ ?

Cevabı ve açıklamayı göster

Cevap: -17

Cevap

The position of point

R

is represented by the integer

-17

.

To find the location of point

R

, we first determine the location of point

Q

by moving

8

units in the positive direction (right) from point

P

(at

-14

), which gives

-14 + 8 = -6

. Next, we find the location of point

R

by moving

11

units in the negative direction (left) from point

Q

(at

-6

), which gives

-6 - 11 = -17

.

Adım Adım Çözüm

1

Calculate the position of point

Q

.

-6

Since point

Q

is

8

units from point

P

(which is at

-14

) in the positive direction, we add

8

to

-14

:

-14 + 8 = -6

.

2

Calculate the position of point

R

.

-17

Since point

R

is

11

units from point

Q

(which is at

-6

) in the negative direction, we subtract

11

from

-6

:

-6 - 11 = -17

.

Anahtar Kavram

Adding and subtracting integers on a number line to find positions relative to a starting point.

Tahmini Süre:45s

Pre-Algebra

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