Examkin

Pre-Algebra

419 questions

All types419Multiple choice291Numeric93Ordering35
Question 61Question →

A positive integer nn has the prime factorization 2a×3b×5c2^a \times 3^b \times 5^c, where aa, bb, and cc are positive integers. The greatest common divisor of nn and 840840 is 120120, and the least common multiple of nn and 9090 is 1,8001,800. If nn is not divisible by 99, what is the value of a+b+ca + b + c?

Show answer & explanation

Answer: 6

Answer

6
The correct answer is 6. By analyzing the prime factorizations, the greatest common divisor condition GCD(n,840)=120\text{GCD}(n, 840) = 120 tells us that the exponents of the prime factors of nn satisfy a≥3a \geq 3, b≥1b \geq 1, and c≥1c \geq 1. The least common multiple condition LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800 tells us that a=3a = 3, b≤2b \leq 2, and c=2c = 2. Finally, the condition that nn is not divisible by 9 requires b<2b < 2, which forces b=1b = 1. Summing these values gives 3+1+2=63 + 1 + 2 = 6.

Step-by-Step Solution

1
Find the prime factorizations of the given numbers.
840=23×31×51×71840 = 2^3 \times 3^1 \times 5^1 \times 7^1, 120=23×31×51120 = 2^3 \times 3^1 \times 5^1, 90=21×32×5190 = 2^1 \times 3^2 \times 5^1, and 1,800=23×32×521,800 = 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 GCD(n,840)=120\text{GCD}(n, 840) = 120, we have min⁡(a,3)=3\min(a, 3) = 3, min⁡(b,1)=1\min(b, 1) = 1, and min⁡(c,1)=1\min(c, 1) = 1. This implies a≥3a \geq 3, b≥1b \geq 1, and c≥1c \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 LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800, we have max⁡(a,1)=3\max(a, 1) = 3, max⁡(b,2)=2\max(b, 2) = 2, and max⁡(c,1)=2\max(c, 1) = 2. This implies a=3a = 3, b≤2b \leq 2, and c=2c = 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=3a = 3 and c=2c = 2. The exponent bb must satisfy 1≤b≤21 \leq b \leq 2. Since nn is not divisible by 9=329 = 3^2, the exponent of 3 in the factorization of nn must be strictly less than 2. Thus, b=1b = 1.
If b=2b = 2, then nn would contain 323^2, making it divisible by 9, which violates the given constraint.
5
Calculate the sum a+b+ca + b + c.
a+b+c=3+1+2=6a + b + c = 3 + 1 + 2 = 6.
Adding the determined exponents yields the final requested value.

Key Concept

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

Alternative Method

Instead of checking each prime factor independently, we can find the value of nn directly. Since GCD(n,840)=120\text{GCD}(n, 840) = 120, nn must be a multiple of 120120. The multiples of 120120 are 120,240,360,480,600,720,840,…120, 240, 360, 480, 600, 720, 840, \dots. We test which of these multiples satisfies LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800 and is not divisible by 99. Since 1,800=15×1201,800 = 15 \times 120, nn must divide 1,8001,800. The divisors of 1,8001,800 that are multiples of 120120 are 120,240,360,600,900,1,800120, 240, 360, 600, 900, 1,800. Among these, only 600600 has a greatest common divisor of 120120 with 840840, is not divisible by 99, and has LCM(600,90)=1,800\text{LCM}(600, 90) = 1,800. Thus, n=600=23×31×52n = 600 = 2^3 \times 3^1 \times 5^2, which gives a+b+c=3+1+2=6a+b+c = 3+1+2 = 6.
Estimated Time:2m 0s
Question 62Question →

A certain type of single-celled organism has a length of approximately 2.5×10−62.5 \times 10^{-6} meters. If 4×1034 \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?

Show answer & explanation

Answer: 1.0×10−21.0 \times 10^{-2}

Answer

The total length of the line is 1.0×10−21.0 \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: (2.5×10−6)×(4×103)(2.5 \times 10^{-6}) \times (4 \times 10^3). Grouping the coefficients and powers of ten gives (2.5×4)×(10−6×103)=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-3}. To write this in standard scientific notation, rewrite 1010 as 1.0×1011.0 \times 10^1, which yields 1.0×101+(−3)=1.0×10−21.0 \times 10^{1 + (-3)} = 1.0 \times 10^{-2}.

Step-by-Step Solution

1
Set up the multiplication of the two values to find the total length.
Total Length =(2.5×10−6)×(4×103)= (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 1010 together, then perform the multiplication.
(2.5×4)×(10−6×103)=10×10−6+3=10×10−3(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: 10a×10b=10a+b10^a \times 10^b = 10^{a+b}.
3
Convert the result into standard scientific notation a×10na \times 10^n where 1≤∣a∣<101 \le |a| < 10.
10×10−3=1.0×101×10−3=1.0×10−210 \times 10^{-3} = 1.0 \times 10^1 \times 10^{-3} = 1.0 \times 10^{-2}
Since 1010 is not less than 1010, rewrite 1010 as 1.0×1011.0 \times 10^1 and add the exponents (1+(−3)=−21 + (-3) = -2) to express the number in standard scientific notation.

Key Concept

Multiplying numbers in scientific notation and converting to standard scientific notation.
Estimated Time:1m 0s
Question 63Question →

A basket contains red apples and green apples in a ratio of 2:52:5. If there are 35 apples in the basket, how many green apples are in the basket?

Show answer & explanation

Answer: 25

Answer

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=72 + 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÷7=535 \div 7 = 5 apples per part. Finally, multiply the number of parts representing green apples (55 parts) by the value of each part (55): 5×5=255 \times 5 = 25. Therefore, the correct quantity of green apples in the basket is 25.

Step-by-Step Solution

1
Find the total number of parts in the ratio.
2+5=72 + 5 = 7 parts
The ratio of red apples to green apples is 2:52:5, so the total quantity is divided into 77 equal parts.
2
Calculate the value of a single part of the ratio.
35÷7=535 \div 7 = 5 apples per part
Divide the total number of apples (3535) by the total number of parts (77) 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 parts×5 apples/part=255 \text{ parts} \times 5 \text{ apples/part} = 25 green apples
Since green apples represent 55 parts of the ratio, multiply 55 by the value of one part (55) to find the total number of green apples.

Key Concept

Solving word problems involving part-to-part ratios and totals by finding the unit value per ratio part.
Question 64Question →

A positive integer KK has exactly 1212 positive factors. The greatest common divisor of KK and 8484 is 66, and the least common multiple of KK and 2424 is 360360. What is the value of KK?

Show answer & explanation

Answer: 90

Answer

90
The prime factorizations are 84=22×3×784 = 2^2 \times 3 \times 7, 6=2×36 = 2 \times 3, 24=23×324 = 2^3 \times 3, and 360=23×32×5360 = 2^3 \times 3^2 \times 5. The condition gcd⁡(K,84)=6\gcd(K, 84) = 6 implies the power of 2 in KK is exactly 1, and the power of 3 is at least 1. The condition lcm(K,24)=360\text{lcm}(K, 24) = 360 implies the power of 3 in KK is exactly 2, and the power of 5 is exactly 1. Thus, KK must be 21×32×51=902^1 \times 3^2 \times 5^1 = 90. Testing 9090, it has (1+1)(2+1)(1+1)=12(1+1)(2+1)(1+1) = 12 positive factors, which perfectly satisfies all conditions.

Step-by-Step Solution

1
Find the prime factorization of the given numbers in the problem.
84=22×3×784 = 2^2 \times 3 \times 7, 6=2×36 = 2 \times 3, 24=23×324 = 2^3 \times 3, and 360=23×32×5360 = 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 KK.
2
Analyze the greatest common divisor constraint gcd⁡(K,84)=6\gcd(K, 84) = 6.
The prime 2 must have an exponent of exactly 1 in the factorization of KK (since 8484 has 222^2 and the greatest common divisor has only 212^1). The prime 3 must have an exponent of at least 1 in KK. The prime 7 cannot be a factor of KK.
The greatest common divisor selects the minimum exponent for each shared prime factor.
3
Analyze the least common multiple constraint lcm(K,24)=360\text{lcm}(K, 24) = 360.
The prime 3 must have an exponent of exactly 2 in KK (since 2424 has 313^1 and the least common multiple has 323^2). The prime 5 must have an exponent of exactly 1 in KK (since 2424 has 505^0 and the least common multiple has 515^1). KK 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 KK and verify its factor count.
K=21×32×51=90K = 2^1 \times 3^2 \times 5^1 = 90. The number of positive factors of 9090 is (1+1)(2+1)(1+1)=2×3×2=12(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.

Key Concept

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

Alternative Method

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.
Estimated Time:2m 30s
Question 65Question →

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?

Show answer & explanation

Answer: 72

Answer

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=22×312 = 2^2 \times 3 and 18=2×3218 = 2 \times 3^2. The LCM is 22×32=4×9=362^2 \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=7236 + 36 = 72.

Step-by-Step Solution

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.

Key Concept

Least Common Multiple (LCM)
Estimated Time:1m 30s
Question 66Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of expressions from least to greatest is the expression evaluating to −25-25, followed by the expression evaluating to −8-8, then the expression evaluating to −4-4, and finally the expression evaluating to 1616.
Evaluating each expression using the correct order of operations yields −25-25 for the expression containing the negative base exponent, −8-8 for the expression containing the absolute value, −4-4 for the expression containing the cubed term, and 1616 for the expression containing the squared term. Ordering these values from least to greatest results in the sequence −25<−8<−4<16-25 < -8 < -4 < 16.

Step-by-Step Solution

1
Evaluate the first expression: 2×(−3)2−42÷82 \times (-3)^2 - 4^2 \div 8.
Value is 1616.
Exponents are calculated first: (−3)2=9(-3)^2 = 9 and 42=164^2 = 16. Then, perform multiplication and division: 2×9=182 \times 9 = 18 and 16÷8=216 \div 8 = 2. Finally, subtract: 18−2=1618 - 2 = 16.
2
Evaluate the second expression: −24+3×(5−8)-2^4 + 3 \times (5 - 8).
Value is −25-25.
Parentheses are evaluated first: 5−8=−35 - 8 = -3. Next, evaluate the exponent: −24=−16-2^4 = -16 (since the exponent applies to the base 22 before negation). Perform multiplication: 3×(−3)=−93 \times (-3) = -9. Finally, add: −16+(−9)=−25-16 + (-9) = -25.
3
Evaluate the third expression: (3−5)3÷4−2(3 - 5)^3 \div 4 - 2.
Value is −4-4.
Parentheses are evaluated first: 3−5=−23 - 5 = -2. Next, cube the result: (−2)3=−8(-2)^3 = -8. Perform division: −8÷4=−2-8 \div 4 = -2. Finally, subtract: −2−2=−4-2 - 2 = -4.
4
Evaluate the fourth expression: −∣−6−22×3∣+10-| -6 - 2^2 \times 3 | + 10.
Value is −8-8.
Within the absolute value bars, evaluate the exponent first: 22=42^2 = 4, then multiply: 4×3=124 \times 3 = 12. Subtract to find the absolute value interior: −6−12=−18-6 - 12 = -18. The absolute value of −18-18 is 1818, which is negated to −18-18. Finally, add 1010 to get −18+10=−8-18 + 10 = -8.
5
Order the final values from least to greatest.
−25<−8<−4<16-25 < -8 < -4 < 16.
Comparing the values −25-25, −8-8, −4-4, and 1616 on the number line determines their relative size.

Key Concept

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.
Estimated Time:1m 30s
Question 67Question →

An expression is shown below.

23×22−4×(1−3)3∣2−23∣−10 \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?

Show answer & explanation

Answer: -16

Answer

The correct answer is −16-16.
To find the value of the expression, follow the standard order of operations (PEMDAS). First, simplify terms with exponents and parentheses: 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. The numerator becomes 8×4−4×(−8)=32−(−32)=648 \times 4 - 4 \times (-8) = 32 - (-32) = 64. In the denominator, evaluate the exponent first to get 2−8=−62 - 8 = -6, then apply the absolute value to get ∣−6∣=6|-6| = 6, and finally subtract 1010 to get 6−10=−46 - 10 = -4. Dividing the numerator by the denominator gives 64−4=−16\frac{64}{-4} = -16.

Step-by-Step Solution

1
Evaluate the exponents in the numerator and the denominator, including the term inside the parenthesis.
In the numerator, 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. In the denominator, 23=82^3 = 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 8×4−4×(−8)=32−(−32)8 \times 4 - 4 \times (-8) = 32 - (-32). The denominator becomes ∣2−8∣−10=∣−6∣−10|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=6432 + 32 = 64. The denominator is 6−10=−46 - 10 = -4.
Subtracting a negative number is equivalent to adding its positive counterpart: 32−(−32)=32+3232 - (-32) = 32 + 32. The absolute value of −6-6 is 66, so the denominator simplifies to 6−10=−46 - 10 = -4.
4
Divide the simplified numerator by the simplified denominator.
64−4=−16\frac{64}{-4} = -16.
The fraction bar represents division, which is the final operation to perform to find the value of the expression.

Key Concept

Order of Operations and Number Properties

Alternative Method

Instead of evaluating 23×222^3 \times 2^2 separately, the laws of exponents can be used to simplify the term first: 23×22=23+2=25=322^3 \times 2^2 = 2^{3+2} = 2^5 = 32.
Estimated Time:1m 30s
Question 68Question →

What is the value of the expression below?

−[(−3)2−∣4−2×5∣]×23−18÷3×(−2)−3+1 -[(-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 −30-30.
Applying the correct order of operations (PEMDAS) ensures the expression is evaluated systematically. Evaluating the first term gives −[(−3)2−∣4−2×5∣]×23=−[9−∣−6∣]×8=−[9−6]×8=−3×8=−24-[(-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 18÷3×(−2)−3+1=6×(−2)−2=−12−2=6\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=−30-24 - 6 = -30.

Step-by-Step Solution

1
Simplify the grouping symbols: the bracket and absolute value term.
−3-3
First, evaluate the exponent (−3)2=9(-3)^2 = 9. Inside the absolute value, multiplication takes precedence over subtraction: 4−2×5=4−10=−64 - 2 \times 5 = 4 - 10 = -6. The absolute value is ∣−6∣=6|-6| = 6. Subtracting this from 99 inside the brackets gives 33. Finally, apply the negative sign in front of the brackets to get −3-3.
2
Evaluate the exponent 232^3 and perform the multiplication.
−24-24
Exponents are evaluated before multiplication. Since 23=82^3 = 8, the multiplication becomes −3×8=−24-3 \times 8 = -24.
3
Evaluate the fraction term.
66
In the numerator, multiplication and division are performed from left to right: 18÷3=618 \div 3 = 6, then 6×(−2)=−126 \times (-2) = -12. The denominator evaluates to −3+1=−2-3 + 1 = -2. Dividing the numerator by the denominator gives −12−2=6\frac{-12}{-2} = 6.
4
Perform the final subtraction.
−30-30
Subtract the evaluated fraction value from the first term: −24−6=−30-24 - 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 69Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is: 6.8×10−56.8 \times 10^{-5}, 7.1×10−47.1 \times 10^{-4}, 1.5×10−31.5 \times 10^{-3}, and 2.4×10−32.4 \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<−3-5 < -4 < -3. This places 6.8×10−56.8 \times 10^{-5} as the smallest and 7.1×10−47.1 \times 10^{-4} as the second smallest. For the remaining two values that share the exponent −3-3, we compare their coefficients: 1.5<2.41.5 < 2.4, placing 1.5×10−31.5 \times 10^{-3} before 2.4×10−32.4 \times 10^{-3}.

Step-by-Step Solution

1
Compare the exponents of the base 10 to establish the primary order of magnitude.
The exponents are −5-5, −4-4, and −3-3. Since −5<−4<−3-5 < -4 < -3, we know that any number with an exponent of −5-5 is smaller than one with −4-4, which in turn is smaller than one with −3-3.
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 6.8×10−56.8 \times 10^{-5}, followed by 7.1×10−47.1 \times 10^{-4}.
These have the smallest exponents (−5-5 and −4-4, respectively).
3
Compare the coefficients of the remaining numbers that share the same exponent.
Both 1.5×10−31.5 \times 10^{-3} and 2.4×10−32.4 \times 10^{-3} share the exponent −3-3. Comparing their coefficients, 1.5<2.41.5 < 2.4, so 1.5×10−3<2.4×10−31.5 \times 10^{-3} < 2.4 \times 10^{-3}.
When the powers of 10 are identical, the values can be compared directly by their coefficients.

Key Concept

Comparing and ordering numbers written in scientific notation by analyzing their powers of 10 and coefficients.
Question 70Question →

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

Show answer & explanation

Answer: 12

Answer

12
The correct answer is 12. To satisfy the given conditions, the product xyz>0xyz > 0 means either all three numbers are positive, or exactly two are negative. Since they sum to −3-3, they cannot all be positive. Letting two be negative and one positive, we find that the integer set {−1,2,−4}\{-1, 2, -4\} satisfies ∣−1∣<∣2∣<∣−4∣|-1| < |2| < |-4| (or 1<2<41 < 2 < 4), has a sum of −3-3, and a product of 8>08 > 0. The sum of the absolute differences is ∣−1−2∣+∣2−(−4)∣+∣−4−(−1)∣=3+6+3=12|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12, which is the minimum possible value.

Step-by-Step Solution

1
Analyze the sign constraints from the product condition xyz>0xyz > 0.
Since the product of xx, yy, and zz 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≥1x, y, z \geq 1. This implies their sum x+y+z≥3x+y+z \geq 3, which contradicts the constraint x+y+z=−3x+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-u and −v-v (with u>v>0u > v > 0) and the positive integer be p>0p > 0. The sum constraint becomes −u−v+p=−3-u - v + p = -3, which simplifies to p=u+v−3p = u + v - 3. The absolute values are uu, vv, and pp, which must be three distinct positive integers satisfying ∣x∣<∣y∣<∣z∣|x| < |y| < |z|.
This translates the constraints into positive integer variables representing the absolute values.
4
Determine the values of u,v,pu, v, p that minimize the maximum difference between any two of the three integers x,y,zx, y, z.
For v=1v = 1: if u=4u = 4, then p=2p = 2. The absolute values {1,2,4}\{1, 2, 4\} are distinct. The corresponding integers are x=−1x = -1, y=2y = 2, and z=−4z = -4. Their sum is −3-3 and product is 8>08 > 0. The difference between the maximum and minimum values is 2−(−4)=62 - (-4) = 6. For v=2v = 2: if u=4u = 4, then p=3p = 3. The absolute values {2,3,4}\{2, 3, 4\} are distinct, giving the integers −4,−2,3-4, -2, 3, with a maximum difference of 3−(−4)=73 - (-4) = 7. For higher values, the difference only increases.
The expression ∣x−y∣+∣y−z∣+∣z−x∣|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=−4x = -1, y = 2, z = -4, the value of the expression is ∣−1−2∣+∣2−(−4)∣+∣−4−(−1)∣=3+6+3=12|-1 - 2| + |2 - (-4)| + |-4 - (-1)| = 3 + 6 + 3 = 12.
This is the smallest possible sum of the absolute differences.

Key Concept

Using absolute values and sign analysis to solve system of constraints on integers and distances on a number line.
Question 71Question →

A positive integer is said to have exactly 88 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=p1a1⋯pkakn = p_1^{a_1} \cdots p_k^{a_k} is (a1+1)⋯(ak+1)=8(a_1+1)\cdots(a_k+1) = 8. The possible factorization shapes are p7p^7 (smallest is 128), p3qp^3 q (smallest are 24, 40, 54, 56, 88), and pqrp 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 (a1+1)(a2+1)⋯(ak+1)=8(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 p7p^7, p3qp^3 q, and pqrp q r for distinct primes pp, qq, and rr.
These correspond to the integer factorizations of 8: 8, 4×24 \times 2, and 2×2×22 \times 2 \times 2.
3
List the smallest candidate values for each structure.
Candidates include 128 (for p7p^7); 24, 40, 54, 56, 88 (for p3qp^3 q); and 30, 42, 66, 70 (for pqrp 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 72Question →

What integer represents the midpoint between −9-9 and 77 on a standard number line?

Show answer & explanation

Answer: −1-1

Answer

The midpoint coordinate is −1-1.
The midpoint of two numbers on a number line is their mathematical average. Adding the coordinates −9-9 and 77 gives a sum of −2-2. Dividing this sum by 22 yields the midpoint coordinate of −1-1.

Step-by-Step Solution

1
Identify the coordinates of the two endpoints on the number line.
The endpoints are −9-9 and 77.
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 a+b2\frac{a + b}{2}.
−9+72=−22=−1\frac{-9 + 7}{2} = \frac{-2}{2} = -1.
The midpoint of any two coordinates on a number line is their arithmetic mean.

Key Concept

Finding the midpoint between two integers on a number line
Estimated Time:45s
Question 73Question →

What is the value of the mathematical expression below?

−18÷3×2−(6−8)3-18 \div 3 \times 2 - (6 - 8)^3
Show answer & explanation

Answer: -4

Answer

-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=−26 - 8 = -2. Second, evaluate the exponent to get (−2)3=−8(-2)^3 = -8. Next, perform multiplication and division from left to right: first divide −18-18 by 33 to get −6-6, then multiply by 22 to get −12-12. Finally, perform the subtraction: −12−(−8)=−12+8=−4-12 - (-8) = -12 + 8 = -4.

Step-by-Step Solution

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÷3=−6-18 \div 3 = -6
Multiplication and division have equal priority and must be performed from left to right.
4
Perform the multiplication.
−6×2=−12-6 \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.

Key Concept

Order of Operations (PEMDAS) with negative numbers and exponents
Estimated Time:1m 0s
Question 74Question →

On a map of a city, the scale is 0.5 inches=12 miles0.5\text{ inches} = 12\text{ miles}. If the actual distance between two towns is 96 miles96\text{ miles}, what is the distance between the two towns on the map, in inches?

Show answer & explanation

Answer: 4

Answer

4 inches
To find the distance on the map, we set up the proportion comparing map inches to actual miles: 0.5 inches12 miles=x inches96 miles\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}}. Cross-multiplying gives 12x=4812x = 48, which simplifies to x=4x = 4. Therefore, the distance on the map is 4 inches4\text{ inches}.

Step-by-Step Solution

1
Set up a proportion to relate the map distance to the actual distance.
0.5 inches12 miles=x inches96 miles\frac{0.5\text{ inches}}{12\text{ miles}} = \frac{x\text{ inches}}{96\text{ miles}} where xx 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 xx by cross-multiplying.
12⋅x=0.5⋅9612 \cdot x = 0.5 \cdot 96
12x=4812x = 48
Cross-multiplication is the standard algebraic method to solve proportions.
3
Divide both sides of the equation by 12 to find xx.
x=4812=4x = \frac{48}{12} = 4
Isolating the variable gives the final map distance.

Key Concept

Solving ratio scale problems by setting up and solving proportions.
Question 75Question →

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

Show answer & explanation

Answer: 36

Answer

There are 36 distinct pairs of integers (x,y)(x, y) that satisfy the inequality.
By analyzing the possible values for ∣x∣|x| and ∣y∣|y| within the set {0,1,2,3,4}\{0, 1, 2, 3, 4\}, we identify the 12 pairs that satisfy the inequality ∣∣x∣−∣y∣∣≥2||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∣|x| and ∣y∣|y| are {0,1,2,3,4}\{0, 1, 2, 3, 4\}.
Since xx and yy are integers satisfying −4≤x,y≤4-4 \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∣)(|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)(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∈{0,1,2,3,4}u, v \in \{0, 1, 2, 3, 4\} such that ∣u−v∣≥2|u - v| \geq 2.
3
Calculate the number of integer coordinate pairs (x,y)(x, y) corresponding to each absolute value pair.
Pairs with one zero value yield 2 integer solutions (e.g., u=0,v=2⇒(0,2),(0,−2)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⇒(1,3),(1,−3),(−1,3),(−1,−3)u=1, v=3 \Rightarrow (1, 3), (1, -3), (-1, 3), (-1, -3)).
An absolute value of 0 corresponds to only 1 integer (00), whereas any positive absolute value kk corresponds to 2 integers (kk and −k-k).
4
Sum the number of integer pairs for all valid cases.
Total pairs = (3×2)+(3×2)+(6×4)=6+6+24=36(3 \times 2) + (3 \times 2) + (6 \times 4) = 6 + 6 + 24 = 36.
There are 3 pairs with u=0u=0 (66 solutions), 3 pairs with v=0v=0 (66 solutions), and 6 pairs with both u,v>0u, v > 0 (2424 solutions).

Key Concept

Solving nested absolute value inequalities with integer constraints and counting solution pairs systematically.
Question 76Question →

A high-speed fiber-optic cable transmits data at a rate of 8.0×1078.0 \times 10^7 bytes per second. A digital archive containing a total of 3.2×10113.2 \times 10^{11} bytes needs to be transmitted. How many seconds will it take to complete the transmission?

Show answer & explanation

Answer: 4.0×1034.0 \times 10^3

Answer

The transmission will take 4.0×1034.0 \times 10^3 seconds.
The correct answer represents the transmission time of 4.0×1034.0 \times 10^3 seconds. This is determined by dividing the total data volume, 3.2×10113.2 \times 10^{11} bytes, by the rate, 8.0×1078.0 \times 10^7 bytes per second. The division yields 3.28.0×1011−7=0.4×104\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 1010 by one, resulting in 4.0×1034.0 \times 10^3.

Step-by-Step Solution

1
Set up the division to find the transmission time.
Time=3.2×1011 bytes8.0×107 bytes/second\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.
3.28.0=0.4\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.
1011107=1011−7=104\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.
0.4×104=4.0×1030.4 \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.

Key Concept

Dividing numbers in scientific notation and converting to standard form
Estimated Time:1m 0s
Question 77Question →

A cleaning mixture is prepared by combining bleach and water in a ratio of 1:81:8. If a container holds a total of 3636 cups of this mixture, how many cups of bleach were used?

Show answer & explanation

Answer: 44

Answer

4 cups
The correct answer is 4. The ratio of bleach to water is 1:81:8, which means the entire mixture is divided into 1+8=91 + 8 = 9 equal parts. Bleach constitutes 19\frac{1}{9} of the overall mixture. By taking 19\frac{1}{9} of the 3636-cup total volume, we find that the amount of bleach used is 44 cups.

Step-by-Step Solution

1
Determine the total parts in the mixture's ratio.
1+8=91 + 8 = 9 total parts
Since the ratio of bleach to water is 1:81:8, the mixture is composed of 11 part bleach for every 88 parts water.
2
Express the bleach portion as a fraction of the total mixture.
19\frac{1}{9}
The bleach represents 11 part out of the 99 total parts in the mixture.
3
Find the amount of bleach in the 3636-cup container.
19×36=4\frac{1}{9} \times 36 = 4 cups
Multiply the fraction of bleach by the total volume of the cleaning mixture.

Key Concept

Determining a part from a part-to-part ratio and a total quantity
Question 78Question →

An artist has a rectangular sheet of stained glass that measures 8484 inches by 120120 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, 8484 and 120120. The prime factorizations are 84=22×3×784 = 2^2 \times 3 \times 7 and 120=23×3×5120 = 2^3 \times 3 \times 5, which gives a GCF of 22×3=122^2 \times 3 = 12 inches. The number of square tiles that fit along the length is 84÷12=784 \div 12 = 7 and along the width is 120÷12=10120 \div 12 = 10. Multiplying these dimensions gives a total of 7×10=707 \times 10 = 70 square tiles.

Step-by-Step Solution

1
Determine the prime factorizations of the dimensions of the sheet.
84=22×3×784 = 2^2 \times 3 \times 7 and 120=23×3×5120 = 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 8484 and 120120.
GCF(84,120)=22×3=12\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÷12=784 \div 12 = 7 tiles along the width, and 120÷12=10120 \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.
7×10=707 \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 79Question →

Two positive integers, aa and bb, are such that a<ba < b. The greatest common divisor of aa and bb is 1212, and their least common multiple is 720720. If aa is a multiple of 55 but bb is not a multiple of 55, what is the value of aa?

Show answer & explanation

Answer: 60

Answer

60
By representing a=12xa = 12x and b=12yb = 12y with gcd⁡(x,y)=1\gcd(x, y) = 1 and x<yx < y, the least common multiple constraint gives 12xy=72012xy = 720, which simplifies to xy=60xy = 60. Since aa is a multiple of 55 and bb is not, the factor of 55 in 6060 must belong to xx. Given that x<yx < y and gcd⁡(x,y)=1\gcd(x, y) = 1, the only valid coprime factorization of 6060 where the factor 55 is in xx and x<yx < y is x=5x = 5 and y=12y = 12. This yields a=12×5=60a = 12 \times 5 = 60.

Step-by-Step Solution

1
Express the two numbers in terms of their greatest common divisor (GCD).
a=12xa = 12x and b=12yb = 12y, where gcd⁡(x,y)=1\gcd(x, y) = 1 and x<yx < y.
Since the greatest common divisor of aa and bb is 1212, both numbers must be multiples of 1212, and their remaining parts xx and yy must be coprime to ensure their GCD is exactly 1212.
2
Use the least common multiple (LCM) to find the product of xx and yy.
xy=60xy = 60.
The least common multiple of 12x12x and 12y12y when gcd⁡(x,y)=1\gcd(x, y) = 1 is 12xy12xy. Setting 12xy=72012xy = 720 and dividing by 1212 gives xy=60xy = 60.
3
Identify the constraints on xx and yy based on the divisibility by 55.
55 must divide xx, and 55 must not divide yy.
We are given that a=12xa = 12x is a multiple of 55, which means 55 must be a factor of xx. Since b=12yb = 12y is not a multiple of 55, 55 cannot be a factor of yy.
4
Determine the unique pair (x,y)(x, y) that satisfies all constraints.
x=5x = 5 and y=12y = 12.
The product xy=60xy = 60 has prime factorization 22×3×52^2 \times 3 \times 5. Since gcd⁡(x,y)=1\gcd(x, y) = 1, the factor 55 must belong to xx, and the other prime factors 222^2 and 33 can be distributed. To satisfy x<yx < y, the only possible assignment is x=5x = 5 and y=12y = 12 (since other assignments like x=15,y=4x = 15, y = 4 or x=20,y=3x = 20, y = 3 violate x<yx < y).
5
Calculate the value of aa.
a=60a = 60.
Since a=12xa = 12x and x=5x = 5, we find a=12×5=60a = 12 \times 5 = 60.

Key Concept

Using prime factorizations to analyze greatest common divisors and least common multiples under algebraic and inequality constraints.
Question 80Question →

Point PP is located at −14-14 on a standard number line. Point QQ is located 88 units from point PP in the positive direction. Point RR is located 1111 units from point QQ in the negative direction. What integer represents the location of point RR?

Show answer & explanation

Answer: -17

Answer

The position of point RR is represented by the integer −17-17.
To find the location of point RR, we first determine the location of point QQ by moving 88 units in the positive direction (right) from point PP (at −14-14), which gives −14+8=−6-14 + 8 = -6. Next, we find the location of point RR by moving 1111 units in the negative direction (left) from point QQ (at −6-6), which gives −6−11=−17-6 - 11 = -17.

Step-by-Step Solution

1
Calculate the position of point QQ.
−6-6
Since point QQ is 88 units from point PP (which is at −14-14) in the positive direction, we add 88 to −14-14: −14+8=−6-14 + 8 = -6.
2
Calculate the position of point RR.
−17-17
Since point RR is 1111 units from point QQ (which is at −6-6) in the negative direction, we subtract 1111 from −6-6: −6−11=−17-6 - 11 = -17.

Key Concept

Adding and subtracting integers on a number line to find positions relative to a starting point.
Estimated Time:45s
PreviousPage 4 / 21Next