TWO INTEGERS ARE DEFINED AS "PARTNERS" IF BOTH OF THEIR PRIME FACTORIZATIONS CONTAIN ALL THE SAME PRIME FACTORS. FOR EXAMPLE, 15 AND 45 ARE PARTNERS SINCE BOTH ARE DIVISIBLE BY THE SAME SET OF PRIME NUMBERS 3 AND 5. HOW MANY POSITIVE INTEGERS GREATER THAN 50 AND LESS THAN 100 HAVE ATLEAST ONE PARTNER GREATER THAN 50 AND LESS THAN 100?
partner pairs (and their factors)
54 and 72 (2 3)
54 and 96 (2 3)
56 and 98 (2 7)
60 and 90 (2 3 5)
72 and 96 (2 3)
To find the number of positive integers greater than 50 and less than 100 that have at least one partner within the same range, we can apply the concept of prime factorization.
Let's break it down step by step:
Step 1: Find the prime factorization of each number within the given range (greater than 50 and less than 100).
Starting with the number 51:
51 = 3 * 17
Next, the number 52:
52 = 2 * 2 * 13
And so on, continuing this process for all the numbers from 51 to 99.
Step 2: Determine the partners for each number within the range.
To determine the partners for each number, we need to compare their prime factorizations. Two numbers are partners if they have the same set of prime factors.
For example, let's consider the number 51. Its prime factorization is 3 * 17. To find its partners, we need to find other numbers within the range whose prime factorizations also include 3 and 17. Checking the list, we find that 51 has no partners within the given range because none of the numbers from 51 to 99 have both 3 and 17 as prime factors.
Similarly, we need to check for each number within the range and see if it has any partners within the same range.
Step 3: Count the numbers that have at least one partner.
Go through the entire range of numbers and count how many have at least one partner within the same range. In this case, we are only interested in the number of distinct numbers that have at least one partner; we don't need to count each partner separately.
Once you have gone through the entire range and identified the numbers that have at least one partner, count the total.
It's important to note that while performing the above steps, we can use various strategies to optimize the process, such as avoiding repetition, eliminating numbers that have already been counted, etc.
Following this process, you can obtain the final count of positive integers greater than 50 and less than 100 that have at least one partner within the same range.