Question

Ernie, Bert, and the Cookie Monster want to measure the length of Sesame Street. Each of them does it his own way. Ernie relates: "I made a chalk mark at the beginning of the street and then again every 7 feet. There were 2 feet between the last mark and the end of the street." Bert tells you: "Every 11 feet, there are lamp posts in the street. The first one is 5 feet from the beginning, and the last one is exactly at the end of the street." Finally, the Cookie Monster says: "Starting at the beginning of Sesame Street, I put down a cookie every 13 feet. I ran out of cookies 22 feet from the end." All three agree that the length does not exceed 1000 feet. How long is Sesame Street?

Short answer

The length of Sesame Street is 319 feet.

Step-by-step solution

Analyze Ernie's Information

Ernie marks every 7 feet until he reaches 2 feet before the end. If the street is \( L \) feet long, then \((L - 2)\) must be divisible by 7. Therefore, we have the equation:\[L = 7k + 2\]for some integer \( k \).

Analyze Bert's Information

Bert places lamp posts every 11 feet, starting at 5 feet with the last at the end. Thus, \( L - 5 \) is divisible by 11. This gives us:\[L = 11m + 5\]for some integer \( m \).

Analyze Cookie Monster's Information

The Cookie Monster places cookies every 13 feet and stops 22 feet before the end. Therefore, \( L - 22 \) is divisible by 13. This leads to the equation:\[L = 13n + 22\]for some integer \( n \).

Solve the System of Equations

To find a common solution for \( L \), we can solve the system of equations:1. \( L = 7k + 2 \)2. \( L = 11m + 5 \)3. \( L = 13n + 22 \)We need to find values of \( k, m, \) and \( n \) such that \( L \leq 1000 \). We can solve these equations by substituting each possibility for \( L \) until all conditions fit. This is done using the Chinese Remainder Theorem or manual trial and error.

Find Common Solutions

We start testing integer values to find a common \( L \). When all conditions are met, and \( L \leq 1000 \), it leads to the value:Testing various values, we determine \[L = 319 \]is consistent with all three sets of conditions.

Key concepts

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a fundamental principle in number theory that allows us to solve systems of equations with different moduli. In this exercise, each friend marks Sesame Street in a particular pattern, which can be described by modular equations:

• Ernie uses a modulus of 7, since his marks fit the pattern every 7 feet.
• Bert uses 11, as lamp posts repeat every 11 feet starting from 5 feet.
• Cookie Monster places cookies every 13 feet, starting from the beginning and missing the last 22 feet.

When you use CRT, you look for a number that fits all these patterns simultaneously.

The CRT asserts that if the moduli are pairwise coprime (which means no two numbers share any factors other than 1), then there is a unique solution modulo the product of these moduli. Here, 7, 11, and 13 are coprime, so CRT guarantees a unique solution for Sesame Street's length modulo 7*11*13.

This simplifies our problem by reducing the number of trials needed to test potential solutions.

Number Theory

Number theory, often known as the "Queen of Mathematics," helps us explore properties and relationships of integers. In the context of this problem, number theory helps us formulate equations based on modular arithmetic.

Each friend's marking spree can be viewed through the lens of number theory:

• Ernie creates a sequence by subtracting 2 feet from the end and mentioning a division by 7.
• Bert's measurements involve placing lamp posts every 11 feet.
• Cookie Monster places cookies at intervals of 13 feet.

Understanding these sequences and overlaps is crucial in determining the length of Sesame Street. These overlapping cycles align when viewed correctly, offering insights into synchronized periods - a concept deeply rooted in number theory.

Thus, number theory was instrumental in deriving and solving the equations that led to finding the street length.

Divisibility

Divisibility is a central concept in number theory and plays a crucial role in solving this problem. It delves into how integers can be divided into equal parts without leaving a remainder, which is key to understanding the problem set by Ernie, Bert, and Cookie Monster.

For Ernie, his reports on marking every 7 feet imply that the length minus 2 needs to be evenly divisible by 7. Bert's placement of lamp posts signifies a similar rule but with 11 feet intervals, starting from 5 feet. Cookie Monster operates under the divisible rule of 13 feet per cookie, minus the final 22 feet.

Understanding divisibility allows us to form equations:

• Ernie: \(L = 7k + 2\)
• Bert: \(L = 11m + 5\)
• Cookie Monster: \(L = 13n + 22\)

These capture the essence of how various lengths divide Sesame Street without issue, leading us to a singular solution via the methods of CRT and number theory. This concept helps us to eliminate values that don't adhere to the rules of divisibility in the problem, streamlining our search for the correct street length.