All practice questions

Question 21Question →

On a standard number line, point $A$ is located at $-7$ and point $B$ is located at $5$ . What is the distance between point $A$ and point $B$ ?

Show answer & explanation

Answer: 12

Answer

The distance between the two points is 12.

The distance between two points on a number line is the absolute value of their difference. Calculating

|-7 - 5|

yields

|-12|

, which is

12

. Alternatively, subtracting the lesser coordinate from the greater coordinate gives

5 - (-7) = 5 + 7 = 12

.

Step-by-Step Solution

1

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

The coordinates are

-7

for point

A

and

5

for point

B

.

These coordinates represent the positions from which we need to find the distance.

2

Apply the absolute value formula for the distance between two points,

|a - b|

.

The expression is

|-7 - 5|

.

Distance on a number line is always non-negative and is defined by the absolute value of the difference between the two coordinates.

3

Perform the subtraction and find the absolute value.

|-12| = 12

.

Subtracting

5

from

-7

gives

-12

, and the absolute value of

-12

is

12

.

Key Concept

The distance between two points

a

and

b

on a number line is given by

|a - b|

.

Estimated Time:45s

Question 22Question →

An algebra student is simplifying the expression $a(b - c) - d(e - f)$ by applying the distributive property. Instead of distributing correctly, the student incorrectly writes the expansion as $a(b - c) - de - df$ . If $a = 2$ , $b = 3$ , $c = 5$ , $d = -3$ , $e = 4$ , and $f = -2$ , what is the absolute difference between the student's incorrect result and the correct value of the expression?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

The correct expression evaluates to 14, and the incorrect expression evaluates to 2. The absolute difference between them is

|14 - 2| = 12

.

Step-by-Step Solution

1

Evaluate the correct expression

a(b - c) - d(e - f)

using the given values.

14

To find the mathematically correct value of the expression.

2

Evaluate the incorrect expression

a(b - c) - de - df

using the same values.

2

To find the value resulting from the student's distribution error.

3

Find the absolute difference between the correct value and the incorrect value.

12

To determine the error magnitude as requested by the question.

Key Concept

Distributive Property and Order of Operations

Question 23Question →

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 24Question →

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 25Question →

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 26Question →

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 27Question →

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 28Question →

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 29Question →

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 30Question →

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 31Question →

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.

Question 32Question →

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}

Show answer & explanation

Answer: -30

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Alternative Method

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

Estimated Time:2m 30s

Question 33Question →

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

Show answer & explanation

Answer: 54

Answer

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.

Step-by-Step Solution

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.

Key Concept

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

Question 34Question →

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 $||x| - |y|| \geq 2$ ?

Show answer & explanation

Answer: 36

Answer

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.

Step-by-Step Solution

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).

Key Concept

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

Question 35Question →

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?

Show answer & explanation

Answer: 70

Answer

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.

Step-by-Step Solution

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.

Key Concept

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

Question 36Question →

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 $a$ ?

Show answer & explanation

Answer: 60

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Question 37Question →

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 $R$ ?

Show answer & explanation

Answer: -17

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Estimated Time:45s

Question 38Question →

What is the integer value of the expression $-4^2 + | -15 \div 3 \times 2 | - (-3)^3$ ?

Show answer & explanation

Answer: 21

Answer

The correct answer is 21.

The correct answer is obtained by strictly following the order of operations (PEMDAS): first evaluating exponents (

-4^2 = -16

and

(-3)^3 = -27

), then performing the operations inside the absolute value from left to right (

-15 \div 3 = -5

followed by

-5 \times 2 = -10

, which simplifies to

|-10| = 10

), and finally executing addition and subtraction from left to right (

-16 + 10 - (-27) = 21

).

Step-by-Step Solution

1

Evaluate the exponent

-4^2

-16

According to the order of operations, the exponent is applied to the base 4 first, and then the negation is applied, yielding

-(4^2) = -16

.

2

Evaluate the absolute value expression

|-15 \div 3 \times 2|

10

Multiplication and division have the same precedence and must be performed from left to right: first

-15 \div 3 = -5

, then

-5 \times 2 = -10

. Taking the absolute value of

-10

yields

10

.

3

Evaluate the exponent

(-3)^3

-27

Since the negative sign is inside the parentheses, the base is

-3

, so

(-3)^3 = (-3) \times (-3) \times (-3) = -27

.

4

Combine the values and simplify

21

Substitute the evaluated parts into the original expression:

-16 + 10 - (-27)

. Simplify by performing addition and subtraction from left to right:

-16 + 10 = -6

, and

-6 - (-27) = -6 + 27 = 21

.

Key Concept

Order of operations (PEMDAS) and absolute value properties

Question 39Question →

What is the value of the expression $5^3 - \sqrt{64}$ ?

Show answer & explanation

Answer: 117

Answer

The correct answer is

117

.

Evaluating

5^3

gives

125

since

5 \times 5 \times 5 = 125

. The square root of

64

is

8

because

8^2 = 64

. Subtracting

8

from

125

yields the correct result of

117

.

Step-by-Step Solution

1

Evaluate the exponential term

5^3

125

An exponent indicates how many times a base is multiplied by itself. Here,

5^3 = 5 \times 5 \times 5 = 125

.

2

Evaluate the radical term

\sqrt{64}

8

The square root of a number is the non-negative value that, when multiplied by itself, equals the original number. Since

8 \times 8 = 64

,

\sqrt{64} = 8

.

3

Subtract the evaluated terms

117

Subtract the value of the radical from the value of the exponent to simplify the expression:

125 - 8 = 117

.

Key Concept

Evaluating basic exponents and square roots

Question 40Question →

If $a = 3$ and $b = 2$ , what is the value of the expression $\sqrt{a^4 - b^5}$ ?

Show answer & explanation

Answer: 7

Answer

The value of the expression is 7.

Evaluating the exponents yields

3^4 = 81

and

2^5 = 32

. Substituting these values under the radical gives

\sqrt{81 - 32} = \sqrt{49}

. Evaluating the square root of 49 gives the correct value of 7.

Step-by-Step Solution

1

Evaluate the terms with exponents using the given values

a = 3

and

b = 2

.

a^4 = 3^4 = 81

and

b^5 = 2^5 = 32

Exponent operations must be evaluated before performing subtraction or taking the root.

2

Substitute the evaluated terms back into the radical expression and subtract.

\sqrt{81 - 32} = \sqrt{49}

Simplifying the expression under the radical is necessary before taking the square root.

3

Find the square root of the simplified value.

\sqrt{49} = 7

Evaluating the square root of 49 completes the simplification of the expression.

Key Concept

Evaluating expressions involving exponents, subtraction, and square roots

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