Hello, young explorers of the numerical realm! Today, we’re embarking on a thrilling quest to uncover the secrets of creating the mightiest and the most minuscule products using the magical digits 1 to 5. Are you ready to unlock the mysteries of mathematics and become a master of digits? Let’s set sail on this mathematical adventure!

Knowledge Points:

  • Understanding maximum and minimum value problems
  • The concept of weight in numbers
  • Strategies for finding the maximum and minimum products

The Quest for the Greatest Product

Imagine we have two brave numbers, $abc$ and $de$, made from the digits 1 to 5. They want to combine their strengths to create the biggest and smallest products possible. But how do they do it? Let’s find out!

Why Weight Doesn’t Work Here

At first glance, you might think that the weight of each digit (how much it’s worth based on its position) will help us solve this puzzle. But there’s a twist! The weight of $a$ in $abc$ is $de00$, and the weight of $d$ in $de$ is $abc0$. It’s like trying to compare the strength of an elephant with the speed of a cheetah – they’re both impressive, but in different ways. So, using weight to find our maximum and minimum just won’t work this time.

The Secret Formula

There’s a magical rule in the land of numbers that says when the sum of two numbers is fixed, the smaller their difference, the bigger their product. But wait! The sum of $abc + de$ isn’t fixed, is it? Let’s put on our wizard hats and transform this problem into one where the sum is constant.

Aligning the Stars, I Mean, Digits!

Let’s line up our numbers like this:

a  b  c
d  e

By doing this, we group the digits by their place value: $a$ and $d$ are like the kings of a 4-digit kingdom ($de00$ and $abc0$), $b$ and $e$ are the queens of a 3-digit realm ($de0$ and $abc$), and $c$ is the prince of a 2-digit territory ($de$). Now, we’re ready to cast our enchanting spell.

Transforming de into de0

What if we add a zero to the end of $de$ to make it $de0$? It’s like giving $de$ a magical cloak that doesn’t change its true nature but makes it easier to compare with $abc$. Now, if $abc \times de0$ is at its maximum, then so is $abc \times de$. And guess what? The sum of $abc + de0$ is constant:

\[\displaylines{ a + d = 4 + 5 (or 5 +4) = 9 \\ b + e = 2 + 3 (or 3 +2) = 5 \\ c + 0 = 1 + 0 (or 0 +1) = 1 }\] \[(abc + de0) = (a+d) \times 100 + (b+e) \times 10 + (c + 0) \times 1 = 900 + 50 + 1 = 951\]

With the sum now fixed, we can use our secret formula to find the maximum product.

Crafting the Largest Product

First, let’s write down the hundreds place for our two numbers:

5 x x  
4 x x

We see that the number above is larger, and the number below is smaller. To maximize the product, we want the larger number to be as small as possible and the smaller number to be as large as possible. Clearly, $5xx > 4xx$, so for the tens place, which holds the numbers 3 and 2, we should give the larger 3 to the smaller number $4xx$:

5 2 x
4 3 x

For the ones place, holding the digits 0 and 1, we again give the larger number 1 to the smaller number $4xx$:

5 2 0
4 3 1

Thus, the solution to our original quest is that the product of $52 \times 431$ is the largest possible.

Crafting the Smallest Product

Using the same method, we can find the smallest value:

1 x x (as small as possible)
2 x x (as large as possible)
=>
1 3 x (as small as possible)
2 4 x (as large as possible)
=>
1 3 0 (as small as possible)
2 4 5 (as large as possible)

Let’s summarize our method for finding the product’s maximum and minimum values:

Align the numbers to the left, for a larger product take the larger number and make it as small as possible, and for a smaller product take the smaller number and make it as small as possible.

Another Example: The Digits 1, 3, 7, 5, 9

Let’s embark on another quest with the digits 1, 3, 7, 5, 9. We want to form a two-digit number and a three-digit number. What are the largest and smallest products we can create?

For the largest product, we choose the largest digits and make the larger number as small as possible:

9
7
=>
9 3
7 5
=>
9 3 0
7 5 1

The largest product is $930 \times 751 = 698430$.

For the smallest product, we choose the smallest digits and make the smaller number as small as possible:

1
3
=>
1 5
3 7
=>
1 5 0
3 7 9

The smallest product is $150 \times 379 = 56850$.

Yet Another Example: The Digits 0, 2, 4, 6, 8

Now, let’s tackle the digits 0, 2, 4, 6, 8. What are the largest and smallest products we can form with a two-digit and a three-digit number?

For the largest product, we choose the largest digits and make the larger number as small as possible:

8
6
=>
8 2
6 4
=>
8 2 0
6 4 0

The largest product is $820 \times 640 = 524800$.

For the smallest product, we choose the smallest digits and make the smaller number as small as possible:

2
4
=>
2 0 
4 6
=>
2 0 0
4 6 8

The smallest product is $200 \times 468 = 93600$.

Conclusion

Align the numbers to the left, for a larger product take the larger number and make it as small as possible, and for a smaller product take the smaller number and make it as small as possible.

In this enchanting journey, we’ve learned that by aligning numbers and strategically placing digits, we can maximize or minimize products. Remember, young wizards of numbers, the key to success is understanding the place value and using it to your advantage. Now, go forth and apply these magical strategies to conquer any numerical challenge that comes your way!