# 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?

### Comments for Product of Positive Integers Adds to 20 - Product of Sum

 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