Let 

𝜏

(

𝑚

,

𝑛

)

=

[

𝑇

(

𝑚

,

𝑛

)

𝑇

(

𝑛

,

𝑚

)

]

τ(m,n)=[ 

T(n,m)

T(m,n)

​

 ], where 

[

𝑥

]

[x] means 

𝑥

x rounded down. What is 

𝜏

(

10

⋅

1

0

9

,

5

⋅

1

0

9

)

τ(10⋅10 

9

 ,5⋅10 

9

 )?

We are given the function \(\tau(m, n) = \left\lfloor \frac{T(m, n)}{T(n, m)} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the floor of \(x\), and \(T(m, n)\) refers to a value related to \(m\) and \(n\) (perhaps it represents some transformation or operation on these two numbers). 

You are asked to compute \(\tau(10 \cdot 10^9, 5 \cdot 10^9)\).

Since we are not explicitly told what the function \(T(m, n)\) is, I'll assume you meant the greatest common divisor (gcd), which is a common choice in such problems. If \(T(m, n) = \text{gcd}(m, n)\), then:

\[
\tau(m, n) = \left\lfloor \frac{\text{gcd}(m, n)}{\text{gcd}(n, m)} \right\rfloor
\]

Since \(\text{gcd}(m, n) = \text{gcd}(n, m)\), this simplifies to:

\[
\tau(m, n) = \left\lfloor 1 \right\rfloor = 1
\]

Thus, the answer is:

\[
\tau(10 \cdot 10^9, 5 \cdot 10^9) = 1
\]

Let me know if you need more clarification or if the function \(T\) refers to something else!

---

Here are five related questions you could explore:

1. What are other possible definitions for the function \(T(m, n)\)?
2. How would \(\tau(m, n)\) change if \(T(m, n)\) were defined as the least common multiple?
3. Can you compute \(\tau(m, n)\) for different values of \(m\) and \(n\)?
4. What is the result of \(\tau(15, 10)\) if \(T(m, n) = m + n\)?
5. What if \(T(m, n)\) were defined as \(m^n\) (i.e., \(m\) raised to the power \(n\))?

**Tip:** Understanding the definition of a function like \(T\) is crucial to solving related mathematical problems. Always confirm function definitions when interpreting mathematical expressions.

Let τ(m,n) = [T(m,n) / T(n,m)], where [x] means x rounded down. What is τ(10⋅10^9, 5⋅10^9)?

Explore how to compute τ(10⋅10^9, 5⋅10^9) using the greatest common divisor (GCD) and the floor function. This problem involves key concepts in number theory and highlights the properties of GCD. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation