together paul & harvey are going to visit their aunt on sunday. If paul visit their aunt every 6 days while harvey every 8 days, on what day will they visit their aunt together?
(With solution pls).. tnx...
the easy way is just to list the sequences of days:
6 12 18 24 ...
8 16 24 ...
and see where they collide.
The general way to do it algebraically is to note that
6 = 2*3
8 = 2^3
The smallest multiple of both 6 and 8 is therefore 2^3 * 3 = 24
To determine the day they will visit their aunt together, we need to find the common multiple of 6 and 8.
Step 1: Find the prime factors of the two numbers:
- Prime factors of 6: 2 x 3
- Prime factors of 8: 2 x 2 x 2
Step 2: Write the prime factors of both numbers side by side, including each factor the greatest number of times it appears in either factorization:
- Common prime factors: 2 x 2 x 2 x 3
Step 3: Multiply all the common prime factors:
- 2 x 2 x 2 x 3 = 24
Step 4: This means that it will take 24 days for Paul and Harvey to visit their aunt together.
If they start counting from Sunday, they will visit their aunt together a total of 24 days later.
Since Sunday is the 1st day, counting 24 days later would be 1 + 24 = 25.
Therefore, they will visit their aunt together on the 25th day, which falls on a Monday.
To find out on what day Paul and Harvey will visit their aunt together, we need to determine the least common multiple (LCM) of the numbers 6 and 8. The LCM is the smallest number that is divisible by both 6 and 8.
To find the LCM, we can calculate the multiples of each number until we find a common multiple. Let's start with Paul, who visits their aunt every 6 days:
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Now let's list the multiples of Harvey, who visits their aunt every 8 days:
Multiples of 8: 8, 16, 24, 32, 40, ...
From the lists above, we see that both Paul and Harvey will visit their aunt together on the 24th day, as it is the earliest common multiple.
Therefore, they will visit their aunt together on the 24th day, which corresponds to Sunday.