Pre-Algebra Practice Questions — ACT — Page 3

Question 41Question →

The prime factorization of a positive integer $N$ is $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If $N$ has exactly $12$ distinct positive factors, what is the least possible value of $N$ ?

Show answer & explanation

Answer: 60

Answer

60

The number of distinct positive factors of

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

is given by the formula

(a+1)(b+1)(c+1)

. Given that

N

has exactly

12

positive factors and

a, b, c

are positive integers (each at least

1

), we must factor

12

into three integers that are each at least

2

. The only way to do this is

2 \times 2 \times 3

. This means the set of exponents

\{a, b, c\}

must be

\{1, 1, 2\}

. To minimize the value of

N

, we pair the largest exponent

2

with the smallest prime base

2

, and the exponents of

1

with the prime bases

3

and

5

. This yields the minimum value

2^2 \times 3^1 \times 5^1 = 60

.

Step-by-Step Solution

1

Determine the formula for the number of positive factors of

N

.

The number of positive factors is

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

.

For any integer expressed as a product of prime powers, the number of positive factors is the product of each exponent increased by 1.

2

Find the possible values for the exponents

a

,

b

, and

c

.

Since

a, b, c \ge 1

, we must have

a+1 \ge 2

,

b+1 \ge 2

, and

c+1 \ge 2

. The only factorization of

12

into three integers each at least

2

is

2 \times 2 \times 3

. Thus, the exponents

\{a, b, c\}

must be

\{1, 1, 2\}

in some order.

We must find a set of integer values for the exponents that satisfy the factor count constraint.

3

Minimize the value of

N

by assigning exponents to the prime bases.

Assign the largest exponent (

2

) to the smallest base (

2

), and the smaller exponents (

1

) to the larger bases (

3

and

5

). This gives

a=2, b=1, c=1

, so

N = 2^2 \times 3^1 \times 5^1 = 60

.

To minimize the total product, the largest power must be applied to the smallest base.

Key Concept

Finding the number of factors of a positive integer using its prime factorization.

Estimated Time:1m 15s

Question 42Question →

What is the value of the expression $-|-15 + 7|$ ?

-22

-15

-8

8

22

Show answer & explanation

Answer: $-8$

Answer

-8

To find the value of

-|-15 + 7|

, we first evaluate the expression inside the absolute value bars:

-15 + 7 = -8

. Next, we take the absolute value of that result:

|-8| = 8

. Finally, we apply the negative sign that is outside the absolute value bars, which gives

-8

.

Step-by-Step Solution

1

Evaluate the expression inside the absolute value bars.

-15 + 7 = -8

Before applying the absolute value, the terms inside the vertical bars must be simplified according to the order of operations.

2

Find the absolute value of the result from Step 1.

|-8| = 8

The absolute value of a number represents its distance from zero on a number line, which is always non-negative.

3

Apply the negative sign that is outside the absolute value bars.

- (8) = -8

The negative sign outside acts as multiplication by

-1

after the absolute value has been evaluated.

Key Concept

Evaluating expressions involving absolute values and negative numbers

Question 43Question →

Let $S$ be the set of all positive integers $n$ such that the least common multiple of $n$ and $40$ is $360$ , and the greatest common divisor of $n$ and $100$ is a prime number. What is the sum of all possible values of $n$ in set $S$ ?

18

45

63

72

153

Show answer & explanation

Answer: 63

Answer

63

To find the sum of all possible values of

n

, we first analyze the prime factorizations:

40 = 2^3 \times 5

and

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

. The least common multiple of

n

and

40

is

360

, meaning

n

must be of the form

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

where

0 \le a \le 3

and

0 \le c \le 1

. Testing the candidate values against the condition that

\text{GCD}(n, 100)

must be a prime number (where

100 = 2^2 \times 5^2

) reveals that only

n = 18

(with GCF of

2

) and

n = 45

(with GCF of

5

) satisfy the conditions. The sum of these values is

18 + 45 = 63

.

Step-by-Step Solution

1

Find the prime factorizations of the given numbers:

40

and

360

.

40 = 2^3 \times 5

and

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

.

Expressing the numbers in their prime factorizations helps determine the required prime factors of

n

to satisfy the least common multiple condition.

2

Determine the constraints on the prime factorization of

n

based on

\text{LCM}(n, 40) = 360

.

n

must be of the form

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

, where

0 \le a \le 3

and

0 \le c \le 1

.

Since the least common multiple is the product of the highest powers of all prime factors present in either number,

n

must provide

3^2

(as

40

has

3^0

), can have

2^a

up to

2^3

, and can have

5^c

up to

5^1

.

3

List all 8 possible candidate values for

n

.

For

c=0

:

n \in \{9, 18, 36, 72\}

. For

c=1

:

n \in \{45, 90, 180, 360\}

.

Evaluating all combinations of the exponents

a

and

c

generates the complete list of candidates.

4

Calculate the greatest common divisor of each candidate

n

with

100 = 2^2 \times 5^2

, and check if the result is prime.

\text{GCD}(9, 100) = 1

(not prime);

\text{GCD}(18, 100) = 2

(prime);

\text{GCD}(36, 100) = 4

(not prime);

\text{GCD}(72, 100) = 4

(not prime);

\text{GCD}(45, 100) = 5

(prime);

\text{GCD}(90, 100) = 10

(not prime);

\text{GCD}(180, 100) = 20

(not prime);

\text{GCD}(360, 100) = 20

(not prime). The valid values for

n

are

18

and

45

.

Only

n = 18

and

n = 45

satisfy the condition that their greatest common divisor with

100

is a prime number (

2

and

5

, respectively).

5

Calculate the sum of the possible values of

n

.

18 + 45 = 63

.

Summing the identified valid values yields the final answer.

Key Concept

Determining possible values of an unknown integer using prime factorization constraints from least common multiple (LCM) and greatest common divisor (GCD) conditions.

Question 44Question →

A positive integer $n$ has exactly two distinct prime factors, $p$ and $q$ , such that $p + q = 12$ . If $n$ has exactly 8 positive factors, what is the least possible value of $n$ ?

35

245

875

1,225

1,715

Show answer & explanation

Answer: 875

Answer

875

The prime factors of the number must be 5 and 7 since they are the only prime numbers that sum to 12. For a number with prime factorization

5^a \times 7^b

, the number of factors is

(a+1)(b+1) = 8

. Since both prime factors must be present, the exponents must be 1 and 3. The two possible numbers are

5^3 \times 7^1 = 875

and

5^1 \times 7^3 = 1,715

. The least value is 875.

Step-by-Step Solution

1

Find the two prime numbers

p

and

q

that sum to 12.

The prime factors are 5 and 7.

Since

p

and

q

are prime and

p + q = 12

, we test pairs of positive integers. The only pair of prime numbers that sums to 12 is 5 and 7.

2

Express the number of factors of

n

using the prime factorization formula.

The number of factors is

(a + 1)(b + 1) = 8

.

For any positive integer

n = p^a \times q^b

, the number of positive factors is given by the formula

(a + 1)(b + 1)

.

3

Determine the possible values for the exponents

a

and

b

.

The set of exponents

\{a, b\}

must be

\{1, 3\}

.

Since

n

must contain both prime factors,

a \ge 1

and

b \ge 1

, which means

(a + 1) \ge 2

and

(b + 1) \ge 2

. The only factor pair of 8 where both factors are at least 2 is

2 \times 4

. Thus, one exponent must be 1 and the other must be 3.

4

Calculate the possible values of

n

and identify the least value.

The least value is 875.

The two possible values for

n

are

5^3 \times 7^1 = 125 \times 7 = 875

and

5^1 \times 7^3 = 5 \times 343 = 1,715

. The least possible value is 875.

Key Concept

Factors, Multiples, and Prime Factorization

Question 45Question →

What is the least common multiple of $15$ and $20$ ?

Show answer & explanation

Answer: 60

Answer

The least common multiple of

15

and

20

is

60

.

The correct answer is

60

. The least common multiple is the smallest positive integer that is divisible by both

15

and

20

. By listing multiples or using prime factorization (

2^2 \times 3 \times 5

), we arrive at

60

.

Step-by-Step Solution

1

Find the prime factorization of

15

and

20

.

15 = 3 \times 5

and

20 = 2^2 \times 5

.

Decomposing numbers into their prime factors makes it straightforward to compute their least common multiple.

2

Identify the highest exponent for each prime factor appearing in the prime factorizations.

The prime factors present are

2

,

3

, and

5

. The highest powers are

2^2

,

3^1

, and

5^1

.

The least common multiple must contain all prime factors of both numbers at their maximum frequency to be divisible by both.

3

Calculate the product of these highest powers of the prime factors.

2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60

.

Multiplying these factors yields the smallest positive integer that is a multiple of both original numbers.

Key Concept

Least Common Multiple (LCM)

Estimated Time:45s

Question 46Question →

Two cyclists, Alan and Beatrice, begin riding laps around a circular track at the same time, starting from the same line and traveling in the same direction. Alan completes one lap every 40 seconds, and Beatrice completes one lap every 50 seconds. What is the number of laps Beatrice will have completed the next time they both cross the starting line at the same instant?

4

5

10

40

200

Show answer & explanation

Answer: 4

Answer

Beatrice will have completed 4 laps.

The correct answer is 4. The two cyclists will meet at the starting line after a period of time that is a multiple of both of their lap times. The first time they meet after starting is the least common multiple (LCM) of 40 and 50, which is 200 seconds. Dividing the total time of 200 seconds by Beatrice's lap time of 50 seconds yields 4 laps.

Step-by-Step Solution

1

Find the prime factorizations of both lap times to determine their least common multiple (LCM).

The prime factorization of 40 is

2^3 \times 5^1

, and the prime factorization of 50 is

2^1 \times 5^2

.

They will cross the starting line together at times that are common multiples of their individual lap times.

2

Calculate the LCM by taking the highest power of each prime factor present in either factorization.

\text{LCM}(40, 50) = 2^3 \times 5^2 = 8 \times 25 = 200

seconds.

The least common multiple represents the minimum amount of time that must pass before both cyclists reach the starting line at the same time.

3

Divide the total time elapsed by Beatrice's time per lap to find the number of laps she completes.

200 \text{ seconds} \div 50 \text{ seconds per lap} = 4

laps.

This converts the total time until they meet into the number of laps completed specifically by Beatrice.

Key Concept

Least Common Multiple (LCM) and rate calculations

Estimated Time:1m 0s

Question 47Question →

If the mathematical expression below is simplified using the standard order of operations, what is the result?

12 - 6 \div 2 \times 3 + 4^2

27

25

11

19

-13

Show answer & explanation

Answer: 19

Answer

The correct result of the expression is 19.

The correct value is obtained by first evaluating the exponent (

4^2 = 16

), then performing the division and multiplication from left to right (

6 \div 2 = 3

, followed by

3 \times 3 = 9

), and finally performing the subtraction and addition from left to right (

12 - 9 = 3

, followed by

3 + 16 = 19

).

Step-by-Step Solution

1

Evaluate the exponent term.

4^2 = 16

Exponents must be evaluated before multiplication, division, addition, and subtraction.

2

Perform multiplication and division from left to right.

6 \div 2 \times 3 = 3 \times 3 = 9

Multiplication and division have equal precedence, so they are processed in order from left to right.

3

Perform addition and subtraction from left to right.

12 - 9 + 16 = 3 + 16 = 19

Addition and subtraction have equal precedence, so they are processed in order from left to right.

Key Concept

Order of Operations (PEMDAS)

Question 48Question →

For all real numbers $a$ and $b$ , which of the following expressions is equivalent to $a - b[a - (b - a)] - b^2$ ?

a

a - 2ab - 2b^2

a - 2ab

2a^2 - 3ab

a - 2ab + 2b^2

Show answer & explanation

Answer: $a - 2ab$

Answer

The expression

a - 2ab

To simplify the expression, we begin inside the innermost grouping symbols (the parentheses) and work outward. First, we distribute the negative sign to simplify the term inside the brackets:

a - (b - a) = a - b + a = 2a - b

. Next, we distribute the

-b

term outside the brackets to get

-b(2a - b) = -2ab + b^2

. Substituting this back into the main expression gives

a - 2ab + b^2 - b^2

. Combining the like terms

b^2 - b^2 = 0

simplifies the expression to

a - 2ab

.

Step-by-Step Solution

1

Simplify the expression inside the brackets by distributing the negative sign across the parentheses.

a - (b - a) = a - b + a = 2a - b

According to the order of operations, terms inside parentheses must be simplified first.

2

Substitute the simplified bracket expression back into the main expression.

a - b[2a - b] - b^2

This sets up the expression for the next operation, which is multiplication.

3

Distribute the term

-b

across the bracketed expression.

-b(2a - b) = -2ab + b^2

Multiplication must be performed before subtraction. Multiplying two negative signs yields a positive term.

4

Combine the distributed terms back into the full expression and simplify like terms.

a - 2ab + b^2 - b^2 = a - 2ab

Combining the positive and negative

b^2

terms results in

0

, leaving the fully simplified expression.

Key Concept

Order of Operations and Number Properties

Alternative Method

Instead of simplifying algebraically, you can substitute simple numbers for

a

and

b

. Let

a = 2

and

b = 3

. Evaluating the original expression:

2 - 3[2 - (3 - 2)] - 3^2 = 2 - 3[2 - 1] - 9 = 2 - 3(1) - 9 = -10

. Substituting

a = 2

and

b = 3

into the correct option

a - 2ab

gives

2 - 2(2)(3) = 2 - 12 = -10

. This numerical substitution confirms the correct algebraic simplification.

Estimated Time:1m 30s

Question 49Question →

What is the value of the mathematical expression below?

-3^2 + 4 \times [18 - 3 \times (8 - 4)] \div 2

Show answer & explanation

Answer: 3

Answer

The correct value of the expression is 3.

Following the standard order of operations, we first simplify the nested parentheses

(8 - 4) = 4

. Next, we evaluate the expression inside the brackets

[18 - 3 \times 4] = [18 - 12] = 6

. We then evaluate the exponent

-3^2 = -9

. We perform the multiplication and division from left to right:

4 \times 6 \div 2 = 12

. Finally, adding

-9 + 12

yields the correct result of 3.

Step-by-Step Solution

1

Simplify the innermost parentheses.

-3^2 + 4 \times [18 - 3 \times 4] \div 2

According to the order of operations (PEMDAS), operations inside parentheses or grouping symbols must be performed first.

2

Simplify the expression inside the square brackets by performing multiplication before subtraction.

-3^2 + 4 \times 6 \div 2

Multiplication has a higher precedence than subtraction.

3

Evaluate the exponent

-3^2

.

-9 + 4 \times 6 \div 2

The exponent applies only to the base 3, not to the negative sign, because there are no parentheses around

-3

.

4

Perform multiplication and division from left to right.

-9 + 12

Multiplication and division have the same precedence and must be evaluated in order from left to right.

5

Add the remaining values.

3

Addition is the final operation to perform.

Key Concept

Order of operations (PEMDAS) dictates the sequence in which operations must be performed to evaluate an expression: Parentheses/Brackets, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Question 50Question →

What is the value of the expression $|-12| - |7 - 15|$ ?

Show answer & explanation

Answer: 4

Answer

The value of the expression is 4.

The absolute value of a number represents its distance from zero on a number line, so

|-12| = 12

. Simplifying inside the second term gives

7 - 15 = -8

, and the absolute value

|-8| = 8

. Subtracting these two results yields

12 - 8 = 4

.

Step-by-Step Solution

1

Evaluate the first term

|-12| = 12

The absolute value of a number is its distance from zero on the number line, which is always non-negative.

2

Simplify the expression inside the second absolute value

7 - 15 = -8

Subtracting 15 from 7 yields a negative difference of -8.

3

Evaluate the second absolute value term

|-8| = 8

The absolute value of -8 is 8.

4

Subtract the two evaluated terms

12 - 8 = 4

Subtracting 8 from 12 gives the final simplified value of 4.

Key Concept

Evaluating expressions with absolute values requires simplifying the terms inside the absolute value grouping symbols first, taking the absolute value of the result, and then performing the subtraction.

Question 51Question →

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. The greatest common divisor of $N$ and $360$ has exactly $12$ positive factors, and the least common multiple of $N$ and $360$ has exactly $72$ positive factors. How many positive factors does $N$ have?

Show answer & explanation

Answer: 36

Answer

The positive integer

N

has exactly 36 positive factors.

By writing the prime factorization of

360

as

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

and expressing the GCD and LCM of

N

and

360

in terms of the minimum and maximum exponents of their prime factors, we establish a system of equations for the exponents. Solving this system yields two cases: either

a = 2

,

b = 1

, and

c = 5

, or

a = 1

with

(b + 1)(c + 1) = 18

. In both cases, the formula for the number of positive factors of

N

, which is

(a + 1)(b + 1)(c + 1)

, evaluates to exactly 36.

Step-by-Step Solution

1

Find the prime factorization of

360

.

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

.

This allows us to write the greatest common divisor (GCD) and least common multiple (LCM) of

N

and

360

in terms of their prime factors.

2

Express the number of factors of

\text{GCD}(N, 360)

using the minimum exponents.

(\min(a, 3) + 1)(\min(b, 2) + 1)(\min(c, 1) + 1) = 12

. Since

c \ge 1

, we have

\min(c, 1) = 1

, which simplifies the equation to

(\min(a, 3) + 1)(\min(b, 2) + 1) = 6

.

The GCD of two numbers is found by taking the minimum of their exponents for each prime factor.

3

Solve for the possible values of

a

and

b

from the GCD factor equation.

Since

a, b \ge 1

, the only integer pairs for the factors of

6

are: Case 1:

a = 2

and

b = 1

, or Case 2:

a = 1

and

b \ge 2

.

We analyze the possible values of

\min(a, 3)

and

\min(b, 2)

that multiply to

6

under the constraints

a, b \ge 1

.

4

Apply the LCM condition to find the number of factors of

N

for both cases.

For Case 1:

\text{LCM}(N, 360)

has factors count

(3+1)(2+1)(c+1) = 72 \implies c = 5

, giving

d(N) = (2+1)(1+1)(5+1) = 36

. For Case 2:

\text{LCM}(N, 360)

has factors count

(3+1)(b+1)(c+1) = 72 \implies (b+1)(c+1) = 18

, giving

d(N) = (1+1)(b+1)(c+1) = 2 \times 18 = 36

.

The LCM uses the maximum of the exponents, which allows us to relate the remaining unknown exponents to the total factor count.

Key Concept

Factors, Multiples, and Prime Factorization

Question 52Question →

For all real numbers $x$ and $y$ , two custom operations, $\oplus$ and $\otimes$ , are defined as follows:

x \oplus y = x + y - 1

x \otimes y = xy - x - y + 2

Which of the following statements about these operations must be true for all real numbers $a$ , $b$ , and $c$ ?

I. $a \oplus (b \oplus c) = (a \oplus b) \oplus c$
II. $a \otimes b = b \otimes a$
III. $a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)$

I only

II only

I and II only

II and III only

I, II, and III

Show answer & explanation

Answer: I, II, and III

Answer

The statement containing all three Roman numerals, I, II, and III, is correct because the associative, commutative, and distributive properties all hold true under these custom definitions.

All three statements are true. First, the operation

\oplus

is associative because both

a \oplus (b \oplus c)

and

(a \oplus b) \oplus c

simplify to

a + b + c - 2

. Second, the operation

\otimes

is commutative because

a \otimes b = ab - a - b + 2

and

b \otimes a = ba - b - a + 2

are algebraically identical. Third, the operation

\otimes

distributes over

\oplus

because both

a \otimes (b \oplus c)

and

(a \otimes b) \oplus (a \otimes c)

simplify to

ab + ac - 2a - b - c + 3

. Therefore, the statement including all three Roman numerals is correct.

Step-by-Step Solution

1

Evaluate Statement I for associativity by expanding

a \oplus (b \oplus c)

and

(a \oplus b) \oplus c

.

Both expressions simplify to

a + b + c - 2

.

Since both groupings yield the same algebraic expression, the operation

\oplus

is associative.

2

Evaluate Statement II for commutativity by comparing

a \otimes b

and

b \otimes a

.

a \otimes b = ab - a - b + 2

and

b \otimes a = ba - b - a + 2

.

Since multiplication and addition of real numbers are commutative (

ab = ba

and

-a - b = -b - a

), the two expressions are equal, meaning the operation

\otimes

is commutative.

3

Evaluate Statement III for distributivity by expanding both sides of

a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)

.

The left side expands to

ab + ac - 2a - b - c + 3

. The right side also expands to

ab + ac - 2a - b - c + 3

.

Because both sides simplify to the exact same expression, the operation

\otimes

distributes over

\oplus

.

Key Concept

Testing the definitions of the commutative, associative, and distributive properties using custom operations.

Question 53Question →

A teacher has a bag of marbles to distribute to a class. If the marbles are divided equally among $8$ students, there are $6$ marbles left over. If the marbles are divided equally among $9$ students, there are $7$ marbles left over. What is the least possible number of marbles in the bag?

Show answer & explanation

Answer: 70

Answer

The least possible number of marbles in the bag is 70.

The correct answer is 70. The problem describes a situation where the number of marbles,

N

, leaves a remainder of

6

when divided by

8

, and a remainder of

7

when divided by

9

. This means

N + 2

is a multiple of both

8

and

9

. The smallest positive integer that is a multiple of both

8

and

9

is their least common multiple, which is

72

. Therefore,

N + 2 = 72

, which gives

N = 70

.

Step-by-Step Solution

1

Analyze the relationship between the divisors and their respective remainders.

For divisor

8

, the remainder is

6

, which is

8 - 6 = 2

less than the divisor. For divisor

9

, the remainder is

7

, which is

9 - 7 = 2

less than the divisor. Therefore, adding

2

to the total number of marbles creates a number that is exactly divisible by both

8

and

9

.

This establishes that the target number plus

2

must be a common multiple of the two divisors.

2

Calculate the least common multiple (LCM) of the divisors

8

and

9

.

The prime factorization of

8

is

2^3

and of

9

is

3^2

. The LCM is the product of the highest powers of all prime factors involved:

2^3 \times 3^2 = 8 \times 9 = 72

.

To find the smallest positive common multiple of

8

and

9

.

3

Subtract

2

from the LCM to find the least possible number of marbles.

72 - 2 = 70

.

Since the bag is

2

marbles short of having a multiple of both

8

and

9

, we subtract

2

from the least common multiple.

Key Concept

Least Common Multiple (LCM) application with remainders

Question 54Question →

A woodworker has two wooden boards. One board is $42$ inches long, and the other is $70$ inches long. She wants to cut both boards into shorter pieces of equal length, with no wood left over. What is the greatest possible length, in inches, of each shorter piece?

2

7

14

28

210

Show answer & explanation

Answer: 14

Answer

14

To find the greatest possible length of the pieces, we need to find the greatest common factor (GCF) of the two board lengths,

42

and

70

. The prime factorization of

42

is

2 \times 3 \times 7

, and the prime factorization of

70

is

2 \times 5 \times 7

. The common prime factors are

2

and

7

. Multiplying these common prime factors gives the GCF:

2 \times 7 = 14

. Therefore, the greatest possible length of each piece is

14

inches.

Step-by-Step Solution

1

Identify the mathematical concept needed to solve the problem.

Since we need to cut two boards of lengths

42

and

70

into equal shorter pieces with no left over wood, we need to find a common factor of

42

and

70

. To find the greatest possible length, we must calculate their Greatest Common Factor (GCF).

This translates the word problem into a clear mathematical operation.

2

Find the prime factorizations of both numbers.

42 = 2 \times 3 \times 7

and

70 = 2 \times 5 \times 7

.

Prime factorization helps systematically identify all factors of the two numbers.

3

Determine the common prime factors and multiply them.

The common prime factors are

2

and

7

. Multiplying them gives

2 \times 7 = 14

.

The GCF is the product of all shared prime factors.

Key Concept

Greatest Common Factor (GCF)

Question 55Question →

Let $f(n)$ represent the number of positive factors of a positive integer $n$ . If $n = 2^x \times 3^y$ , where $x$ and $y$ are positive integers such that $x + y = 7$ , what is the maximum possible value of $f(n)$ ?

Show answer & explanation

Answer: 20

Answer

The maximum possible value of

f(n)

is

20

.

The number of positive factors of

n = 2^x \times 3^y

is

(x + 1)(y + 1)

. Since

x

and

y

are positive integers that sum to 7, the possible pairs for

(x, y)

are

(1, 6)

,

(2, 5)

, and

(3, 4)

(and their reversals). The products

(x+1)(y+1)

for these pairs are

2 \times 7 = 14

,

3 \times 6 = 18

, and

4 \times 5 = 20

. The maximum value is 20.

Step-by-Step Solution

1

Determine the formula for the number of factors of

n

.

f(n) = (x + 1)(y + 1)

For any positive integer expressed in its prime factorization

p^a \times q^b

, the total number of positive factors is

(a + 1)(b + 1)

.

2

Identify the possible values for

x

and

y

given the constraints.

(x, y) \in \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\}

The exponents

x

and

y

must be positive integers (

x, y \geq 1

) that sum to 7.

3

Calculate

(x + 1)(y + 1)

for each pair to find the maximum product.

The maximum product is

4 \times 5 = 20

.

Evaluating the products:

2 \times 7 = 14

,

3 \times 6 = 18

, and

4 \times 5 = 20

. The product of two integers with a fixed sum is maximized when the integers are as close as possible.

Key Concept

Calculating the number of positive factors of an integer from its prime factorization and maximizing the count under exponent constraints.

Estimated Time:1m 30s

Question 56Question →

The temperature at the top of a mountain is $-14^\circ\text{C}$ , and the temperature at the base of the mountain is $8^\circ\text{C}$ . What is the absolute difference, in degrees Celsius, between these two temperatures?

-22

-6

6

14

22

Show answer & explanation

Answer: 22

Answer

22

The correct answer is 22 because the absolute difference between

8

and

-14

is

|8 - (-14)| = |8 + 14| = |22| = 22

. This represents the total distance of 22 units between the two temperatures on a number line.

Step-by-Step Solution

1

Translate the problem into a mathematical expression using absolute value for the difference between the two temperatures.

The absolute difference is given by

|8 - (-14)|

or

|-14 - 8|

.

The absolute difference between two numbers

a

and

b

on a number line is defined as

|a - b|

.

2

Simplify the subtraction inside the absolute value.

8 - (-14) = 8 + 14 = 22

(or

-14 - 8 = -22

).

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

3

Evaluate the absolute value.

|22| = 22

(or

|-22| = 22

).

The absolute value of a number represents its non-negative distance from zero.

Key Concept

Integers, Absolute Value, and Number Lines

Estimated Time:45s

Question 57Question →

Four mathematical expressions are shown below:

* Expression 1: $-3^2 - 12 \div 3 \times 2$
* Expression 2: $-| -4 - 2 \times 3 | + (-2)^3$
* Expression 3: $4 - 3(2 - 3^2)$
* Expression 4: $-2^4 \div 8 \times (-1 - 1)^2$

Order the expressions by their simplified numerical values from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the expressions from least to greatest value is Expression 2, Expression 1, Expression 4, and Expression 3.

Evaluating each expression using the correct order of operations yields: Expression 1 equals

-17

, Expression 2 equals

-18

, Expression 3 equals

25

, and Expression 4 equals

-8

. Arranging these values from least to greatest results in

-18 < -17 < -8 < 25

, corresponding to the sequence: Expression 2, Expression 1, Expression 4, Expression 3.

Step-by-Step Solution

1

Evaluate Expression 1.

-17

Apply the order of operations: first evaluate the exponent

-3^2 = -9

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

12 \div 3 = 4

, and

4 \times 2 = 8

. Finally, perform the subtraction:

-9 - 8 = -17

.

2

Evaluate Expression 2.

-18

Simplify inside the absolute value first by performing multiplication before subtraction:

2 \times 3 = 6

, and

-4 - 6 = -10

. The absolute value of

-10

is

10

, which becomes

-10

due to the negative sign outside. Next, evaluate the exponent:

(-2)^3 = -8

. Finally, add the terms:

-10 + (-8) = -18

.

3

Evaluate Expression 3.

25

Simplify inside the parentheses first, evaluating the exponent before subtracting:

3^2 = 9

, and

2 - 9 = -7

. Next, perform the multiplication:

3 \times (-7) = -21

. Finally, subtract:

4 - (-21) = 4 + 21 = 25

.

4

Evaluate Expression 4.

-8

Simplify the expression inside the parentheses:

-1 - 1 = -2

. Next, evaluate the exponents:

-2^4 = -16

and

(-2)^2 = 4

. Perform division and multiplication from left to right:

-16 \div 8 = -2

, and

-2 \times 4 = -8

.

5

Compare the simplified values to order them from least to greatest.

-18 < -17 < -8 < 25

Comparing the results shows that the values in ascending order are

-18

(Expression 2),

-17

(Expression 1),

-8

(Expression 4), and

25

(Expression 3).

Key Concept

Order of Operations and Number Properties

Estimated Time:2m 30s

Question 58Question →

Four distinct integers, $p$ , $q$ , $r$ , and $s$ , are represented on a standard number line. The distance between $p$ and $q$ is 3, the distance between $q$ and $r$ is 4, and the distance between $r$ and $s$ is 5. What is the minimum possible distance between $p$ and $s$ ?

Show answer & explanation

Answer: 2

Answer

The minimum possible distance between

p

and

s

is 2.

By setting

p = 0

as a reference point, the coordinate for

q

must be either

3

or

-3

. Assuming

q = 3

by symmetry,

r

must be either

-1

or

7

because it is at a distance of 4 from

q

. From

r = 7

, a distance of 5 leads to

s

being at

2

or

12

. From

r = -1

, a distance of 5 leads to

s

being at

4

or

-6

. All of these positions result in four distinct integers. The distances between

p

and

s

are the absolute values of their coordinates, which are

12

,

2

,

6

, and

4

. The minimum of these distances is

2

.

Step-by-Step Solution

1

Establish a coordinate system on the number line.

Let

p

be at position 0. Since the distance between

p

and

q

is 3,

q

is at either

3

or

-3

. By symmetry, we assume

q = 3

.

Setting one point at the origin simplifies the relative distance calculations for all other points.

2

Find the possible coordinates of

r

.

Since the distance between

q

and

r

is 4,

r

is at

3 - 4 = -1

or

3 + 4 = 7

.

The absolute value equation

|q - r| = 4

has two solutions for

r

given

q = 3

.

3

Find the possible coordinates of

s

for each case of

r

.

If

r = 7

, then

s

can be at

7 - 5 = 2

or

7 + 5 = 12

. If

r = -1

, then

s

can be at

-1 - 5 = -6

or

-1 + 5 = 4

. All generated sets contain distinct values, satisfying the requirement.

The absolute value equation

|r - s| = 5

has two solutions for

s

for each candidate coordinate of

r

.

4

Determine the minimum distance between

p

and

s

.

The possible values for the distance

|p - s|

are

|0 - 2| = 2

,

|0 - 12| = 12

,

|0 - (-6)| = 6

, and

|0 - 4| = 4

. The minimum value is 2.

Comparing all possible valid configurations ensures we find the true minimum distance.

Key Concept

Representing distances between points on a number line using absolute values and resolving configurations for distinct integers.

Question 59Question →

What is the value of the following expression?

-3^2 - \frac{16 - 4 \times 5}{2^3 - 10}

7

-10

-11

21

39

Show answer & explanation

Answer: $-11$

Answer

-11

The correct value is

-11

. By following the standard order of operations, we first evaluate the leading term

-3^2 = -9

. Next, we evaluate the numerator of the fraction (

16 - 4 \times 5 = 16 - 20 = -4

) and the denominator of the fraction (

2^3 - 10 = 8 - 10 = -2

). Simplifying the fraction gives

\frac{-4}{-2} = 2

. Finally, subtracting this value from the leading term yields

-9 - 2 = -11

.

Step-by-Step Solution

1

Evaluate the exponent term on the left side of the subtraction.

-3^2 = -(3 \times 3) = -9

According to the order of operations, the exponent applies only to the base

3

, not the negative sign, unless parentheses are present.

2

Evaluate the numerator of the fraction by performing multiplication before subtraction.

16 - 4 \times 5 = 16 - 20 = -4

Multiplication has a higher priority than subtraction in the standard order of operations.

3

Evaluate the denominator of the fraction by evaluating the exponent before subtraction.

2^3 - 10 = 8 - 10 = -2

Exponentiation must be performed before subtraction.

4

Divide the simplified numerator by the simplified denominator to find the value of the fraction.

\frac{-4}{-2} = 2

Dividing a negative number by a negative number yields a positive result.

5

Perform the final subtraction using the values from Step 1 and Step 4.

-9 - 2 = -11

Subtracting

2

from

-9

moves further left on the number line, resulting in

-11

.

Key Concept

Order of Operations (PEMDAS/BODMAS)

Question 60Question →

If $x = -6$ , what is the value of the expression $12 - |x - 3|$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is 3.

Substituting

x = -6

into the expression

12 - |x - 3|

gives

12 - |-6 - 3|

. Simplifying the subtraction inside the absolute value gives

12 - |-9|

. Since the absolute value of

-9

is

9

, the expression simplifies to

12 - 9

, which equals

3

.

Step-by-Step Solution

1

Substitute

x = -6

into the expression.

12 - |-6 - 3|

Substitute the given value of the variable into the algebraic expression.

2

Simplify the expression inside the absolute value symbol.

12 - |-9|

Subtracting 3 from

-6

results in

-9

.

3

Calculate the absolute value.

12 - 9

The absolute value represents the distance from zero on the number line, so

|-9| = 9

.

4

Subtract the numbers to find the final value.

3

12 - 9 = 3

.

Key Concept

Evaluating algebraic expressions involving integers and absolute values.

Pre-Algebra

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