- 初二数学设x,y是非负整数,x+2y是5的倍数,x+y是3的倍数
- 设x,y是非负整数,x+2y是5的倍数,x+y是3的倍数,且2x+y≥99,求7x+5y的最小值.
- 解：设x+2y＝5m，x+y＝3n
x,y是非负整数 ===>5m≥3n ,m≥3n/5
10(x+y)=30n
(7x+5y)+(x+2y)+2(x+y)=30n
7x+5y+5m=24n
7x+5y=24n-5m .......(1)
m≥3n/5===>
7x+5y≥21n ................(2)
2x+y≥99 =====>6x+3y≥297
7x+5y=6x+3y+x+2y≥297+5m
===>24n-5m≥297+5m
===>24n≥297+10m≥297+10*(3n/5) =297+6n
===>18n≥297
n≥16.5 n是整数即n≥17
m≥3n/5 ,是整数===>m≥11
n=17,m=11
x+y=51,x+2y=55 ====>x=47,y=4,此时2x+y=98不合题目意思
n=18,m=12
x+y=54,x+2y=60====>x=48,y=6,此时2x+y=102符合题目意思
此时7x+5y最小值48*7+6*5=366