1.(n+5)^2-(n-1)^2
=[(n+5)+(n-1)][(n+5)-(n-1)]
=(2n+4)*6
=12(n+2)
n为正整数.
所以(n+5)^2-(n-1)^2的值一定被12整除.
2..4m^2+12m+25+9n^2-24n
=4m^2+12m+9+9n^2-24n+16
=(2m+3)^2+(3n-4)^2.
因为(2m+3)^2与(3n-4)^2.均非负,所以.4m^2+12m+25+9n^2-24n的值是非负数.
3.9a^2-12ab+8b^2-4bc+2c^2-4c+4=0
9a^2-12ab+4b^2+4b^2-4bc+c^2+c^2-4c+4=0
(3a-2b)^2+(2b-c)^2+(c-2)^2=0
因为(3a-2b)^2,(2b-c)^2,(c-2)^2均非负,所以
(3a-2b)^2=0,(2b-c)^2=0,(c-2)^2=0
3a-2b=0,2b-c=0,c-2=0.
所以a=2/3,b=1,c=2.