Find the largest three-digit number that can be written in the form 3^m + 2^n, where m and n are positive integers.
3^m goes over 3 digits at 3^7
so m<7
let m = 6 to get 3^6 = 729
so that leaves 270 for 2^n
I know 2^8 = 256 , using 2^9 would put me over.
3^6 + 2^8 = 729+256 = 985
Since $3^7 > 1000$, we know that we only have to test 1 through 6 for $m$. For each value of $m$ from 1 through 5, the greatest possible $n$ is $n=9$, and the largest three-digit number we get from these is $3^5 + 2^9 = 243 + 512=755$. But when $m=6$, the largest three-digit number we get is $3^6 + 2^8 = 729 + 256 = \boxed{985}$.
To find the largest three-digit number that can be written in the form 3^m + 2^n, we can start by finding the highest possible values of m and n.
Since we are looking for a three-digit number, we need 3^m + 2^n to be greater than or equal to 100 but less than 1000.
Let's start by considering the largest possible value for m, which is 6 (since 3^7 is already greater than 1000).
We can now find the value of 3^6 = 729.
Next, let's consider the largest possible value for n. Since 2^n will be added to the 729, we want the largest value of n such that 729 + 2^n is less than 1000.
Let's start with n = 9. In this case, 2^9 = 512, and 729 + 512 = 1241, which is greater than 1000.
Now, let's try n = 8. In this case, 2^8 = 256, and 729 + 256 = 985, which is less than 1000. This is the largest three-digit number that can be written in the form 3^m + 2^n.
Therefore, the largest three-digit number that can be written in the form 3^m + 2^n is 985.
To find the largest three-digit number that can be written in the given form 3^m + 2^n, we can start by experimenting with different values of m and n.
Let's begin by finding the largest possible value for m. We notice that as the value of m increases, 3^m grows exponentially. Since we want a three-digit number, we can estimate the largest possible value of m by evaluating 3^m until we exceed 1000.
By trying different values of m, we can see that 3^6 is equal to 729, which is less than 1000, while 3^7 is equal to 2187, which exceeds 1000. Hence, we can conclude that the largest possible value for m is 6.
Now, let's find the largest possible value for n. Similarly, we'll evaluate 2^n for different values of n until we find the largest power of 2 that, when combined with 3^m, gives a three-digit number.
By trying different values of n, we notice that 2^9 is equal to 512, which is less than 1000, while 2^10 is equal to 1024, which exceeds 1000. Therefore, the largest possible value for n is 9.
Now that we have determined the maximum values for m and n, we can calculate the largest three-digit number that satisfies the condition 3^m + 2^n.
Substituting m = 6 and n = 9 into the equation, we have:
3^6 + 2^9 = 729 + 512 = 1241
After our calculations, we find that 1241 is the largest three-digit number that can be written in the form 3^m + 2^n, where m and n are positive integers.
Oh, numbers and math! I'll do my best to help you out, but brace yourself for a little mathematical clownery!
So, let's see. We want to find the largest three-digit number that can be written in the form 3^m + 2^n, where m and n are positive integers. Now, we could try testing different values of m and n, but that would take forever and this clown has better things to do!
Let's use a little trickery. We know that the largest three-digit number is 999. So, if we can find values of m and n such that 3^m + 2^n = 999, then we've found our answer!
Now, let me put on my clown thinking cap... 🤡
If we start with the number 999 and subtract 2, we get 997. Hmmm... 997 is a prime number. That means we can't express it as the sum of any powers of 3 or 2.
Okay, let's try subtracting 2 again. We get 995. Oh, guess what? It's another prime number! So, no luck there.
One more time. If we subtract 2 from 995, we get 993. Woohoo! A number that's not prime! 🎉
Now, let's see if we can express 993 as the sum of powers of 3 and 2. Let me juggle some numbers in my mind... 🤹♂️
If we rewrite 993 as 3^5 + 2^9, we hit the jackpot! The largest three-digit number that can be written in the form 3^m + 2^n is indeed 993.
Voila! Mission accomplished! I hope my clownish mathematical antics have entertained you along the way. If you have any more questions, feel free to ask!