Question

If $t$ is the sum of three consecutive positive integers, is $t$ a multiple of $24?$ (1) The smallest of the 3 integers is even. (2) $t$ is a multiple of 3

Short answer

No, the sum of the three consecutive positive integers with the given properties is not necessarily a multiple of 24.

Step-by-step solution

Introduce the problem

Given three consecutive positive integers, let these integers be represented as $n,n+1,n+2$. Their sum, $t$, is the focus. We need to determine if $t$ is a multiple of $24$. The statements given will help analyze this.

Sum of three consecutive integers

Calculate the sum of the three consecutive integers: $t=n+(n+1)+(n+2)$. Simplify this to get:$$t=3n+3$$ Factor out the 3: $$t=3(n+1)$$.

Evaluate Statement (1)

Evaluate the first statement: The smallest of the three integers is even. If $n$ is even, then let $n=2k$ for some integer $k$. Substitute into the sum equation: $$t=3(2k+1)$$ $$t=6k+3$$. This does not provide sufficient information to conclude if $t$ is a multiple of 24.

Evaluate Statement (2)

Evaluate the second statement: $t$ is a multiple of 3. From the sum equation, we already deduced $$t=3(n+1)$$.Therefore, $t$ as a multiple of 3 is always true. Again, this statement alone does not determine if $t$ is a multiple of 24.

Analyze both statements combined

Combine both statements: The smallest integer is even and $t$ is a multiple of 3. From step 3, $$t=6k+3$$.To check if this can be a multiple of 24, solve $$6k+3=24m$$ for some integer $m$. Dividing everything by 3 yields $$2k+1=8m$$.This equation implies $2k+1$ must be a multiple of 8, but arbitration shows this is not generally satisfied for integers. Hence combined statements do not assert $t$ is a multiple of 24 as always true.

Key concepts

consecutive integers

Consecutive integers are numbers that follow each other in order. This means there is no gap between the numbers when listed sequentially. For example, 1, 2, and 3 are consecutive integers because each number is one more than the previous number. When we deal with problems involving consecutive integers, it’s helpful to represent them algebraically as:

• First integer: n
• Second integer: n + 1
• Third integer: n + 2

The sum of three consecutive integers can therefore be expressed as:

• Sum: t = n + (n+1) + (n+2)

By simplifying this sum, we get:

• t = 3n + 3

This expression helps in analyzing whether the sum has certain divisibility properties, such as being a multiple of another number.

multiple of 24

To determine if a number is a multiple of 24, we need it to be divisible by both 3 and 8. This is because 24 can be factored into its prime components:

• 24 = 3 × 8

From the given problem, the sum of three consecutive integers simplifies to

• t = 3(n + 1)

The '3' in this equation shows that t is already a multiple of 3. However, to also be a multiple of 8, the term (n + 1) must be a multiple of 8. Without additional information that guarantees (n + 1) meets this requirement, we cannot conclude t is a multiple of 24.

integer properties

Understanding the properties of integers, especially their evenness or oddness, plays a crucial role in solving integer-related problems. An integer n can be classified as:

• Even: If n is divisible by 2
• Odd: If n is not divisible by 2

Consider the problem stating, 'The smallest of the three integers is even.' This guides our representation:

• If n is even, let n = 2k for some integer k

Thus, the sum t becomes:

• t = 3(2k + 1) = 6k + 3

Analyzing whether this can be a multiple of 24 involves checking for factors and suitability to the given conditions.

mathematical reasoning

Mathematical reasoning involves logical thinking to solve problems. It requires assessing given conditions and deriving new insights. In this problem, reasoning is applied in steps:

• Understanding the sum of consecutive integers
• Using properties of multiples
• Evaluating given statements individually and then combining them

Statement (1) told us n is even, but without guaranteeing divisibility by 24.
Statement (2) affirmed divisibility by 3, already known. Together they are analyzed:

• From t = 6k + 3, we derive: 6k + 3 = 24m
• Dividing by 3: 2k + 1 = 8m

Concluding whether this multiple condition holds reliably requires inspecting constraints on n, verifying if combined statements suffice to establish a pattern ensuring divisibility by 24 — which in this problem's context did not universally align.