Problem 54 - Basic College Mathematics

Question

Find the prime factorization of each number. Use divisibility tests where applicable. A passenger in a taxicab asks for the driver's company number. The driver says abruptly, "Sure-you can have my number. Work it out: If you divide it by 2,3 , \(4,5,\) or \(6,\) you will get a remainder of 1 . If you divide it by 11 , the remainder will be 0 , and no driver has a company number that meets these requirements and is smaller than this one." Determine the number.

Step-by-Step Solution

Answer

The number is 121.

Step 1: Understanding the Problem

The problem states that the number, when divided by 2, 3, 4, 5, or 6, leaves a remainder of 1. Additionally, when divided by 11, the remainder is 0. This means the number is a multiple of 11 and can be written in the form of 11k.

Step 2: Setting up the Modular Equations

We need to find a number N such that: N ≡ 1 (mod 2)N ≡ 1 (mod 3)N ≡ 1 (mod 4)N ≡ 1 (mod 5)N ≡ 1 (mod 6)N ≡ 0 (mod 11)

Step 3: Least Common Multiple (LCM)

Since N leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6, we need the LCM of these numbers plus 1. The LCM of 2, 3, 4, 5, and 6 is 60, so N satisfies 60k + 1. Thus, N must be of the form 60k + 1.

Step 4: Combine the Conditions

The number must also be a multiple of 11, implying it must be a multiple of the LCM (660) of 60 and 11, plus 1 (for the least N). Therefore, we check for the smallest k such that 60k + 1 is divisible by 11.

Step 5: Solving for the Smallest Possible Value

We need to find the smallest integer value of k such that (60k + 1) ≡ 0 (mod 11). Trying values:For k = 1: 60(1) + 1 = 61 (not divisible by 11)For k = 2: 60(2) + 1 = 121 (divisible by 11)

Step 6: Conclusion

Therefore, the smallest number N that satisfies all these conditions is 121.

Key Concepts

divisibility tests

Divisibility tests help determine if one number can be evenly divided by another without leaving a remainder. These shortcuts are especially useful when working with large numbers. For instance, a number is divisible by 2 if its last digit is even. For 3, a number is divisible if the sum of its digits is divisible by 3. By applying these tests, we can efficiently check divisibility without performing full division each time. In our problem, we consistently noticed that dividing by 2, 3, 4, 5, or 6 left a remainder of 1. This unique pattern helps in narrowing down our search for the correct number. These repeated remainders prompted us to find the smallest multiple that fit the condition.

least common multiple

The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of them. Calculating the LCM is vital for solving problems where you need numbers that fit multiple constraints simultaneously. For example, in our exercise, the LCM of 2, 3, 4, 5, and 6 is 60. This helped us because any number that fits all these divisibility conditions can be written as 60k + 1. This formula helped simplify the problem and narrow down possible solutions.
To calculate an LCM, you need to find the prime factorization of each number, then take the highest power of each prime that appears in any of the factorizations and multiply them together. The LCM of multiple numbers ensures all original conditions are satisfied simultaneously.

modular arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value known as the modulus. This is like the remainder operation in division but with a structured approach. For example, 'N ≡ 1 (mod 2)' means that N leaves a remainder of 1 when divided by 2. This concept is essential in our problem, which involves finding a number that leaves specific remainders when divided by several different numbers. Using modular arithmetic can simplify the problem-solving process as it helps to keep track of remainders efficiently.
In this exercise, modular arithmetic indicated that the number we sought needed to meet several conditions: it needed to be a multiple of 11 and take on specific remainders when divided by 2, 3, 4, 5, and 6. By converting these conditions into modular equations, we were able to find the solution more systematically.

problem-solving steps

Problem-solving in mathematics often involves breaking a complex problem into simpler, more manageable steps. Here's a step-by-step approach that was applied in this exercise:

• Understand the Problem: Clearly define what is being asked. We determined that the number must have specific properties when divided by various other numbers.
• Set Up the Equations: Represent the conditions using modular arithmetic. We wrote equations to show that N left a remainder of 1 when divided by several numbers and 0 by another.
• Find the LCM: Use the least common multiple to simplify the problem. By calculating the LCM of 2, 3, 4, 5, and 6, we found 60, which helped in narrowing down potential solutions.
• Combine Conditions: Ensure that all conditions are met simultaneously. This required us to check if additional constraints, such as being divisible by 11, were satisfied.
• Solve for the Smallest Possible Value: Test small values to find the smallest number that fits all conditions. We tested small k-values to find that 121 was the smallest number that met all requirements.