If the sum of 1! + 2! + 3! + 4! + ... + 49! + 50! is divided by 15, what is the remainder?
Every number above 14! will have zero remainder, since 15 is one of its factors. So, ask the same question about 1! + 2! + 3! + ...14!
All terms greater than 5! have both 3 and 5 as factors, so ask the same question about 1! + 2! + 3! + 4!
That sum is 33. Dividing that by 15 gives 3 with a remainder of 3.
The remainer of the full sum up to 50!divided by 15 will also be 3.
To find the remainder when the sum of factorials from 1 to 50 is divided by 15, we need to calculate the sum of the factorials and then divide it by 15.
Let's break down the problem step by step:
Step 1: Calculate the factorials
We'll calculate the factorials of the numbers from 1 to 50.
1! = 1
2! = 2
3! = 6
4! = 24
...
47! = 2.5852017e+59
48! = 1.241391e+61
49! = 6.0828186e+62
50! = 3.0414093e+64
Step 2: Calculate the sum of the factorials
Now we'll sum up the factorials from 1 to 50.
1! + 2! + 3! + 4! + ... + 49! + 50!
= 1 + 2 + 6 + 24 + ... + 6.0828186e+62 + 3.0414093e+64
The exact sum would involve writing out all the digits, but for simplicity, let's just consider the remainder when dividing by 15.
Step 3: Find the remainder when divided by 15
To find the remainder, we'll divide the sum of the factorials by 15.
(1 + 2 + 6 + 24 + ... + 6.0828186e+62 + 3.0414093e+64) % 15
Let's evaluate this expression by programming.
Please note that calculating the exact sum and remainder for large values like 50! may not be practical without using external tools, so it's better to use a programming language or calculator for precise calculations.
To find the remainder when the sum of factorials from 1 to 50 is divided by 15, we need to calculate the sum first.
First, let's understand what factorials are. The factorial of a number is the product of that number and all positive integers smaller than it. For example, 4! (4 factorial) is calculated as 4 x 3 x 2 x 1 = 24.
Now, let's calculate the sum of the factorials from 1 to 50:
1! + 2! + 3! + 4! + ... + 49! + 50!
To calculate the sum, we can start by finding the factorial of each number from 1 to 50 and then adding them all together.
If we list the factorials of the numbers from 1 to 50 and add them up, we get a really large number that is not practical to compute manually.
Fortunately, to find the remainder when a large number is divided by another number (in this case, 15), we can use a property called the Remainder Theorem, which states that if a number is divided by another number (m) and the remainder is (r), then any multiple of (m) will leave the same remainder (r) when divided by (m).
In this case, we don't need to calculate the actual sum to find the remainder when it is divided by 15.
Let's observe the pattern between the factorials and the remainders when divided by 15:
1! = 1, remainder = 1
2! = 2, remainder = 2
3! = 6, remainder = 6
4! = 24, remainder = 9
5! = 120, remainder = 0
6! = 720, remainder = 0
...
14! = a really large number, remainder = 0
15! = a really large number, remainder = 0
From the pattern we observe, starting from 5!, all factorials are divisible by 15, so their remainders are all 0.
Therefore, the sum of 1! + 2! + 3! + 4! + ... + 49! + 50! is equal to 1 + 2 + 6 + 9 + 0 + 0 + ... + 0, since the sum of the remaining factorials is divisible by 15.
This simplifies the calculation to 1 + 2 + 6 + 9 = 18.
Now, let's calculate the remainder when 18 is divided by 15:
18 divided by 15 gives a quotient of 1 and a remainder of 3.
Therefore, the remainder when the sum of 1! + 2! + 3! + 4! + ... + 49! + 50! is divided by 15 is 3.