GCD and LCM for 756

The greatest common divisor of positive integers m and n is 6. The least common multiple of m and n is 126. What is the least possible value of m + n?

Question: The greatest common divisor of positive integers ( m ) and ( n ) is 6. The least common multiple of ( m ) and ( n ) is 126. What is the least possible value of ( m + n )?

Chain of Thought for Solving the Math Problem:

Understanding the Problem: The problem involves finding the least possible sum of two positive integers ( m ) and ( n ) given that their greatest common divisor (GCD) is 6 and their least common multiple (LCM) is 126. We need to use our understanding of GCDs and LCMs to solve this.

Identifying Known and Unknown Variables: Known variables are the GCD (6) and LCM (126) of ( m ) and ( n ). The unknown variables are ( m ) and ( n ) themselves.

Choosing the Right Method: To solve this, we should use the relationship between the GCD and LCM of two numbers. The product of the GCD and LCM of two numbers equals the product of the numbers themselves, i.e., ( \text{GCD}(m, n) \times \text{LCM}(m, n) = m \times n ).

Setting Up the Problem: Given that ( \text{GCD}(m, n) = 6 ) and ( \text{LCM}(m, n) = 126 ), we can set up the equation ( 6 \times 126 = m \times n ).

Performing Calculations:

• First, calculate the product: ( 6 \times 126 = 756 ). So, ( m \times n = 756 ).
• Now, we need to find pairs of factors of 756 that also have a GCD of 6. Start with the smallest possible factors (other than 6 itself, since we are looking for the least sum of ( m ) and ( n )) and check their GCD.

Checking the Work: Each pair of factors considered must satisfy both the product (756) and the GCD (6) conditions.

Interpreting the Result: The pair of numbers that meets these conditions and yields the smallest possible sum will be our answer.

Let's perform these calculations.

The pairs of factors of 756 that have a GCD of 6 are (6, 126) and (18, 42). Now, let's calculate the sum for each pair to find the least possible value of ( m + n ).

1. For the pair (6, 126), the sum is ( 6 + 126 = 132 ).
2. For the pair (18, 42), the sum is ( 18 + 42 = 60 ).

Among these, the smallest sum is ( 60 ) for the pair (18, 42). Therefore, the least possible value of ( m + n ) is 60.

What is the smallest positive integer solution to (100,000,000/x^4 ) < 1?

Question: What is the smallest positive integer solution to ((100,000,000/x^4 ) < 1)?

Chain of Thought for Solving the Math Problem:

Understanding the Problem: We are asked to find the smallest positive integer ( x ) that makes the inequality ((100,000,000/x^4 ) < 1) true.

Identifying Known and Unknown Variables: The known variable is the constant ( 100,000,000 ) in the inequality. The unknown variable is ( x ), the integer we need to find.

Choosing the Right Method: This is an inequality problem involving division and exponents. We need to manipulate the inequality to isolate ( x ).

Setting Up the Problem: The inequality is ( 100,000,000/x^4 < 1 ). We need to solve for ( x ).

Performing Calculations:

1. First, rewrite the inequality: ( 100,000,000 < x^4 ).
2. Then, find the fourth root of ( 100,000,000 ) to solve for ( x ).
3. Since we are looking for the smallest positive integer solution, we will consider the smallest integer greater than the fourth root of ( 100,000,000 ).

Checking the Work: Ensure the calculated value of ( x ) indeed makes the original inequality true.

Interpreting the Result: The smallest integer ( x ) that satisfies the inequality will be our answer.

Let's perform these calculations.

The fourth root of 100,000,000 is exactly 100. However, since we are looking for the smallest positive integer solution that makes the inequality ((100,000,000/x^4 ) < 1) true, we need an integer greater than 100.

Therefore, the smallest positive integer solution to ((100,000,000/x^4 ) < 1) is ( x = 101 ). This is because ( x = 100 ) would make the inequality an equality, and we need ( x ) to make it strictly less than 1.

Suppose I have a cabbage, a goat and a lion, and I need to get them across a river. I have a boat that can only carry myself and a single other item. I am not allowed to leave the cabbage and lion alone together, and I am not allowed to leave the lion and goat alone together. How can I safely get all three across?

Question: How can I safely get a cabbage, a goat, and a lion across a river using a boat that can carry myself and one other item, without leaving the cabbage and lion alone together or the lion and goat alone together?

Chain of Thought for Solving the Math Problem:

Understanding the Problem: This is a classic river crossing puzzle. The challenge is to figure out a sequence of crossings that respects the constraints about not leaving certain pairs alone together.

Identifying Known and Unknown Variables: The known variables are the entities involved (cabbage, goat, lion, myself) and the constraints. The unknown variable is the sequence of steps to get all across safely.

Choosing the Right Method: This problem requires logical reasoning and the ability to sequence actions while respecting constraints.

Setting Up the Problem: The goal is to get all three items (cabbage, goat, lion) and myself to the other side of the river without ever leaving the cabbage alone with the lion or the lion alone with the goat.

Performing Calculations:

• We'll need to consider the order of transporting items and possibly bringing items back across the river to avoid leaving the wrong pairs together.

Checking the Work: After each step, we need to check if the constraints are violated.

Interpreting the Result: The correct sequence of steps that gets everything across without breaking any rules is our solution.

Let's plan out the steps:

1. Take the lion across first and leave it on the other side.
2. Go back alone and take the goat across.
3. Bring the lion back with you.
4. Take the cabbage across and leave it with the goat.
5. Go back alone to get the lion.

This sequence ensures that the cabbage and lion are not left alone together without you, and the lion and goat are not left alone together without you. Each trip respects the boat's capacity of carrying only one item (or animal) besides yourself.

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Question: Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Chain of Thought for Solving the Math Problem:

Understanding the Problem: This problem involves finding the sum of all positive integers that cannot be expressed as the sum of two abundant numbers. An abundant number is one where the sum of its proper divisors is greater than the number itself.

Identifying Known and Unknown Variables: Known variables include the definition of abundant numbers and the fact that all integers greater than 28123 can be written as the sum of two abundant numbers. The unknown is the list of numbers that cannot be expressed as the sum of two abundant numbers and their sum.

Choosing the Right Method: This problem requires a combination of mathematical analysis and computational approach. We need to:

1. Identify all abundant numbers up to 28123.
2. Determine which numbers up to 28123 cannot be written as the sum of two abundant numbers.
3. Sum these numbers.

Setting Up the Problem: We need to create a list of abundant numbers and then check each number up to 28123 to see if it can be expressed as the sum of two numbers from this list.

Performing Calculations:

1. Find all abundant numbers up to 28123.
2. For each number up to 28123, check if it can be written as the sum of two abundant numbers.
3. Keep track of numbers that cannot be expressed this way and sum them.

Checking the Work: Ensure that the identified numbers truly cannot be written as the sum of two abundant numbers.

Interpreting the Result: The final sum of these numbers will be the answer.

Given the computational nature of this task, I will use programming to find and sum the numbers. Let's perform these steps.

The sum of all positive integers which cannot be written as the sum of two abundant numbers is 4,179,871.

The following iterative sequence is defined for the set of positive integers:

n n/2 (n is even) n 3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence: 13 40 20 10 5 16 8 4 2 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.)

Question: Which starting number, under one million, produces the longest Collatz sequence chain?

Chain of Thought for Solving the Math Problem:

Understanding the Problem: The Collatz sequence is defined as follows: if ( n ) is even, the next number in the sequence is ( n/2 ); if ( n ) is odd, the next number is ( 3n + 1 ). The sequence ends when it reaches 1. The task is to find the starting number below one million that results in the longest sequence.

Identifying Known and Unknown Variables: The known variables are the rules for generating the Collatz sequence. The unknown is the starting number below one million that produces the longest sequence.

Choosing the Right Method: This problem requires a computational approach. We need to generate Collatz sequences for each starting number under one million and keep track of the length of each sequence.

Setting Up the Problem: We will create a function to generate a Collatz sequence from a given starting number and count the terms in the sequence. Then we will apply this function to each number under one million to find the one with the longest sequence.

Performing Calculations:

1. Create a function to generate a Collatz sequence and count its length.
2. Iterate over all numbers from 1 to 999,999.
3. Keep track of the number with the longest sequence and its length.

Checking the Work: Ensure that the sequences are generated correctly according to the Collatz rules.

Interpreting the Result: The number that produces the longest sequence will be our answer.

Let's implement this approach.

The starting number under one million that produces the longest Collatz sequence chain is 837,799, with a sequence length of 525 terms.

You are given a string containing digits 0-9 only.

Write a script to find out if the given string is additive number. An additive number is a string whose digits can form an additive sequence.

A valid additive sequence should contain at least 3 numbers. Except the first 2 numbers, each subsequent number in the sequence must be the sum of the preceding two.

Example 1:

Input: $string = "112358" Output: true

The additive sequence can be created using the given string digits: 1,1,2,3,5,8 1 + 1 => 2 1 + 2 => 3 2 + 3 => 5 3 + 5 => 8

Example 2:

Input: $string = "12345" Output: false

No additive sequence can be created using the given string digits.

Example 3:

Input: $string = "199100199" Output: true

The additive sequence can be created using the given string digits: 1,99,100,199 1 + 99 => 100 99 + 100 => 199

Question: Write a script to determine if a given string of digits is an additive number, where an additive number is defined as a string whose digits can form an additive sequence. An additive sequence is one where each number (after the first two) is the sum of the two preceding numbers.

Chain of Thought for Solving the Math Problem:

Understanding the Problem: The task is to create a function that takes a string of digits and determines if it can be divided into at least three numbers that form an additive sequence.

Identifying Known and Unknown Variables: The known variable is the input string of digits. The unknown is whether this string can be segmented into an additive sequence.

Choosing the Right Method: This problem involves iterating through the string and trying different splits to see if a valid sequence can be formed. We need to check for every possible pair of first two numbers if the rest of the string can form an additive sequence.

Setting Up the Problem: Write a function that iterates over the string, attempting to split it into segments that form an additive sequence.

Performing Calculations:

1. Iterate through the string, choosing different lengths for the first and second number.
2. For each pair of first and second numbers, check if the remaining string forms a valid additive sequence.
3. If any valid sequence is found, return true. If no valid sequences are found, return false.

Checking the Work: Ensure the function correctly identifies valid and invalid additive sequences.

Interpreting the Result: The function should return true if the string is an additive number and false otherwise.

Let's implement this function.

The script correctly identifies whether the given strings are additive numbers:

1. For the input "112358", the output is true. This string can form the additive sequence 1, 1, 2, 3, 5, 8.
2. For the input "12345", the output is false. No additive sequence can be created using these digits.
3. For the input "199100199", the output is true. This string can form the additive sequence 1, 99, 100, 199.

Change the program to also output the numbers that are found.

For example: "112358" should output [1,1,2,3,5,8]. "12345" should output "No solution"

The modified script now outputs the sequences if the string is an additive number, or "No solution" if it's not:

1. For the input "112358", the sequence is [1, 1, 2, 3, 5, 8].
2. For the input "12345", the output is "No solution" as no additive sequence can be formed.
3. For the input "199100199", the sequence is [1, 99, 100, 199].

Please run the program for the following input: 9918274572117189306495801129620973393549088831437323256376296088598514159399

For the input "9918274572117189306495801129620973393549088831437323256376296088598514159399", the sequence formed is: [ [9, 9, 18, 27, 45, 72, 117, 189, 306, 495, 801, 1296, 2097, 3393, 5490, 8883, 14373, 23256, 37629, 60885, 98514, 159399] ]