Highest product

Using the digits 1 to 9, create two numbers which when multiplied together give you the highest number.

For example, 12345678 * 9 = 111111102. Clearly there are higher products. What is the highest?

-Ray.

Could you elaborate? In what way is 111111102 the highest number?

Originally posted by piderman
Could you elaborate? In what way is 111111102 the highest number?

The original text does not state that 111111102 is the highest, but rather that there are higher products. The problem is to find the highest such product. 🙂

-Ray.

OK. I understand now. 97531*8642=842862902. Try to beat that!

Originally posted by piderman
OK. I understand now. 97531*8642=842862902. Try to beat that!

Believe it or not, there is a product that yields a slightly-larger result.

-Ray.

Originally posted by rgoudie
Believe it or not, there is a product that yields a slightly-larger result.

-Ray.

9642*87531 = 843973902

Originally posted by Acolyte
9642*87531 = 843973902

You got it.

-Ray.

I see a trend here, give solution without the derivation...i wish to learn how one solves such a problem

Originally posted by TheMaster37
I see a trend here, give solution without the derivation...i wish to learn how one solves such a problem

I apologize.

In my future postings, I will include a solution if it is not supplied by the person answering.

-Ray.

I'll try to explain:

we have to make abcd * efghi as large as possible

Multiplying this out and collecting together the powers of 10 this is equal to:
10,000,000ae
+1,000,000(be + af)
+100,000(ag + bf + ce)
+10,000(ah + bg + cf + de)
+1,000(ai + bh + cg + df)
+100(bi + ch + dg)
+10(ci + dh)
+ di

Considering the largest (most important) multiple of 10 we see that ae governs its multiplier value, therefore a and e must be 8 and 9, however a appears in more powers of 10 than e, so we will get the biggest answer if a=9, e=8. ie answer = 9bcd * 8fghi

Next most important power of 10 is 1,000,000(8b + 9f) we can see that 7 and 6 must come in here and we get the largest multiplier by multiplying 8*6, 9*7 i.e b=6, f=7, now we have 96cd * 87ghi

Next power of 10 is 100,000(9g + 42 + 8c), 5 and 4 must come in here and this will be largest if g=5, c=4, now we have 964d * 875hi

Considering d and i, it is clear that these have the least impact, so must be where we allocate 1 and 2. i affects powers of 10 up to 1,000, whereas d affects them up to 10,000 so it is best if d=2, i=1: now we have 96c2 * 87gh1

Next smallest numbers (3 and 4) look like they must go in c and h, c affects higher numbers than h, so c=4, h=3 will yield the largest product: 9642 * 87g31

Filling in the missing 5: 9642 * 87531 = 843973902 is the largest product