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Product of Positive Integers Adds to 20 - Product of Sum

by Andrew
(New York)










































If a bunch of positive integers adds up to 20, what is the greatest possible product of these numbers?

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Apr 05, 2012
Product of Positive Integers Adds to 20
by: Staff


Question:

by Andrew
(New York)

If a bunch of positive integers adds up to 20, what is the greatest possible product of these numbers?


Answer:

I used simple trial and error (rather than partition theory)


19+1 = 20, 19*1=19
18+2 = 20, 18*2=36
17+3=20, 17*3=51
16+4=20, 16*4=64
15+5=20, 15*5=75
14+6=20, 14*6=84
13+7=20, 13*7=91
12+8=20, 12*8=96
11+9=20, 11*9=99
10+10=20, 10*10=100

Since order does not matter, this is the highest number we can get adding only 2 natural numbers

10+10=20, 10*10=100
10+9+1=20, 10*9*1=90
10+8+2=20, 10*8*2=160
10+7+3=20, 10*7*3=210
10+6+4=20, 10*6*4=240
10+5+5=20, 10*5*5=250


10+5+5=20, 10*5*5=250
10+5+4+1=20, 10*5*4*1=200
10+5+3+2=20, 10*5*3*2=300


10+5+3+2=20, 10*5*3*2=300
10+4+1+3+2=20, 10*4*1*3*2=240
10+3+2+3+2=20, 10*3*2*3*2=360


10+3+2+3+2=20, 10*3*2*3*2=360
9+1+3+2+3+2=20, 9*1*3*2*3*2=324
8+2+3+2+3+2=20, 8*2*3*2*3*2=576
7+3+3+2+3+2=20, 7*3*3*2*3*2=756
6+4+3+2+3+2=20, 6*4*3*2*3*2=864
5+5+3+2+3+2=20, 5*5*3*2*3*2=900

5+5+3+2+3+2=20, 5*5*3*2*3*2=900
5+3+2+3+3+2+2=20, 5*3*2*3*3*2*2=1080

5+3+2+3+3+2+2=20, 5*3*2*3*3*2*2=1080
3+2+3+2+3+3+2+2=20, 3*2*3*2*3*3*2*2=1296



3+2+3+2+3+3+2+2=20, 3*2*3*2*3*3*2*2=1296

3+3+3+3+3+3+2=20, 3*3*3*3*3*3*2=1458


>>> the greatest product = 1458







Thanks for writing.

Staff
www.solving-math-problems.com



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