Hard ACT Number Properties Practice Questions

Mastering Hard ACT Number Properties Practice Questions is essential for students aiming for a top-tier score on the math section of the ACT Prep journey. These questions often test the underlying logic of integers, remainders, and divisibility rather than simple computation. By understanding the behavior of numbers under various operations, you can solve complex problems that appear in the final third of the exam.

Concept Explanation

Number properties on the ACT refer to the inherent characteristics of integers, rational numbers, and real numbers, including rules for parity, divisibility, prime factorization, and remainders. At a high level, these concepts require you to generalize rules for sets of numbers rather than solving for a specific variable. For instance, you must know that the product of any three consecutive integers is always divisible by 6, or that an even number raised to any positive integer power remains even. Understanding the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 is either a prime or a product of primes, is vital for tackling advanced problems involving factors and multiples.

Key sub-topics include:

• Parity: Rules for adding and multiplying even and odd numbers (e.g., $\text{Odd} \times \text{Odd} = \text{Odd}$).
• Divisibility Rules: Quick ways to determine if a number is divisible by 3, 4, 6, 8, or 9.
• Remainders: Calculating what is left over after division, often used in pattern-based sequence problems.
• Absolute Value: The distance of a number from zero on a number line, which is always non-negative.
• Prime Factors: Breaking numbers down into their prime components to find the Greatest Common Factor (GCF) or Least Common Multiple (LCM).

To prepare effectively, you can use our AI Question Generator to create custom drills that focus specifically on these abstract concepts. This type of practice helps you recognize patterns quickly on test day.

Solved Examples

Example 1: If $n$ is an integer and $n^2$ is divisible by 72, what is the smallest possible positive value of $n$?

1. Find the prime factorization of 72: $72 = 2^3 \times 3^2$.
2. Since $n^2$ is a perfect square, all exponents in its prime factorization must be even.
3. To make $n^2$ divisible by $2^3 \times 3^2$, the prime factorization of $n^2$ must contain at least $2^4 \times 3^2$ (the next highest even power for 2).
4. If $n^2 = 2^4 \times 3^2$, then $n = \sqrt{2^4 \times 3^2} = 2^2 \times 3^1 = 12$.
5. The smallest positive integer $n$ is 12.

Example 2: When the positive integer $x$ is divided by 11, the remainder is 4. What is the remainder when $3x + 5$ is divided by 11?

1. Express $x$ in terms of the quotient $q$: $x = 11q + 4$.
2. Substitute this into the expression: $3(11q + 4) + 5$.
3. Simplify: $33q + 12 + 5 = 33q + 17$.
4. Divide the result by 11: $\frac{33q + 17}{11} = 3q + \frac{17}{11}$.
5. Since 11 goes into 17 once with a remainder of 6, the final remainder is 6.

Example 3: If $a$ and $b$ are positive integers such that $a - b$ is even and $a \times b$ is even, which of the following must be true: $a$ is even, $b$ is even, or both?

1. Analyze the condition $a - b = \text{Even}$. This happens if both $a$ and $b$ are even, or both are odd.
2. Analyze the condition $a \times b = \text{Even}$. This happens if at least one of the numbers is even.
3. Combine the conditions: To satisfy both, the only possibility is that both $a$ and $b$ must be even. If both were odd, their product would be odd, failing the second condition.

Practice Questions

1. If $x$ and $y$ are positive integers such that $\text{GCF}(x, y) = 15$ and $\text{LCM}(x, y) = 180$, and $x = 45$, what is the value of $y$?
2. For how many integer values of $k$ is the expression $\frac{24}{k+1}$ an integer?
3. If $p$ is a prime number greater than 3, what is the remainder when $p^2$ is divided by 12? (Hint: Test small prime numbers like 5 and 7).

4. The set $S$ contains all integers $n$ such that $100 \leq n \leq 200$ and $n$ is a multiple of both 6 and 8. How many integers are in set $S$?
5. If $m$ is an odd integer and $n$ is an even integer, which of the following expressions must be odd: $(m+1)^2$, $m^2 + n$, $m \times n + 1$, or $n/2$?
6. What is the units digit of $3^{43}$?
7. If $a, b,$ and $c$ are consecutive even integers such that $a < b < c$, what is the value of $(c - a)(c - b)(b - a)$?
8. If $x$ is the least common multiple of 12, 18, and 30, and $y$ is the greatest common factor of 45, 75, and 105, what is the value of $x/y$?

Answers & Explanations

1. Answer: 60
Use the property that for any two numbers $x$ and $y$, $x \times y = \text{GCF}(x, y) \times \text{LCM}(x, y)$. Substituting the given values: $45 \times y = 15 \times 180$. Divide both sides by 45: $y = \frac{15 \times 180}{45}$. Since $180/45 = 4$, $y = 15 \times 4 = 60$. For more on relating different math concepts, check our guide on ACT Math Practice Questions with Answers.

2. Answer: 16
The expression is an integer if $k+1$ is a factor of 24. The factors of 24 (both positive and negative) are: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$. There are 8 positive factors and 8 negative factors, totaling 16 possible values for $k+1$. Each value of $k+1$ corresponds to a unique integer $k$.

3. Answer: 1
Test $p = 5$: $5^2 = 25$. $25 \div 12 = 2$ with a remainder of 1. Test $p = 7$: $7^2 = 49$. $49 \div 12 = 4$ with a remainder of 1. Test $p = 11$: $11^2 = 121$. $121 \div 12 = 10$ with a remainder of 1. This is a consistent property for primes greater than 3.

4. Answer: 4
A number is a multiple of both 6 and 8 if it is a multiple of their LCM. $\text{LCM}(6, 8) = 24$. We need to find multiples of 24 between 100 and 200. $24 \times 4 = 96$ (too small), $24 \times 5 = 120$, $24 \times 6 = 144$, $24 \times 7 = 168$, $24 \times 8 = 192$. There are 4 such integers.

5. Answer: $m^2 + n$ and $m \times n + 1$
Let's evaluate: $m^2$ is odd (odd $\times$ odd), and $n$ is even. $\text{Odd} + \text{Even} = \text{Odd}$. Also, $m \times n$ is even (any integer $\times$ even), and $\text{Even} + 1 = \text{Odd}$. Both are always odd. Note that $(m+1)^2$ would be even because $\text{Odd} + 1 = \text{Even}$. If you struggle with these logic-based questions, reviewing ACT Algebra Practice Questions can help clarify variable behavior.

6. Answer: 7
Look for a pattern in the units digits of powers of 3: $3^1=3, 3^2=9, 3^3=27 (7), 3^4=81 (1)$. The pattern $\{3, 9, 7, 1\}$ repeats every 4 powers. Divide the exponent 43 by 4: $43 \div 4 = 10$ with a remainder of 3. The 3rd number in the pattern is 7.

7. Answer: 16
Let the integers be $x, x+2, x+4$. Then $a=x, b=x+2, c=x+4$. Calculate the differences: $(c-a) = 4$, $(c-b) = 2$, $(b-a) = 2$. The product is $4 \times 2 \times 2 = 16$. This remains true regardless of the value of $x$.

8. Answer: 12
Factorize: $12=2^2 \cdot 3$, $18=2 \cdot 3^2$, $30=2 \cdot 3 \cdot 5$. $\text{LCM}(x) = 2^2 \cdot 3^2 \cdot 5 = 180$. For $y$: $45=3^2 \cdot 5$, $75=3 \cdot 5^2$, $105=3 \cdot 5 \cdot 7$. $\text{GCF}(y) = 3 \cdot 5 = 15$. $x/y = 180/15 = 12$.

1. Which of the following must be an even integer if \( x \) is an odd integer?

• A\( x^2 + 2 \)
• B\( 3x + 1 \)
• C\( x/2 \)
• D\( x^3 \)

Frequently Asked Questions

What are number properties on the ACT?

Number properties are rules that describe how different types of numbers—such as integers, odd/even numbers, and primes—interact during mathematical operations. They form the logical foundation for many complex ACT math problems, particularly those involving divisibility and factors.

How do I find the units digit of a large exponent?

To find the units digit of a large exponent, identify the repeating pattern of units digits for the base number's powers. Divide the exponent by the length of the cycle and use the remainder to determine the position in the pattern.

What is the difference between a factor and a multiple?

A factor is a number that divides into another number evenly without a remainder, whereas a multiple is the product of a number and an integer. For example, 3 is a factor of 12, and 12 is a multiple of 3.

Is 1 considered a prime number on the ACT?

No, the number 1 is not considered prime because a prime number must have exactly two distinct positive divisors: 1 and itself. Since 1 only has one divisor, it is excluded from the set of prime numbers.

How do I handle remainder questions with variables?

The most effective way to handle remainder questions with variables is to pick a concrete number that fits the description. For example, if a number divided by 5 leaves a remainder of 2, you can use the number 7 to test the answer choices.