1. (40 points) Prove or disprove each statement. (a) For all integers n1,2n has n/3 digits in base 10 . (b) For all integers x and y, if (x2+1)(y+1) is even then x is odd or y is odd. (c) For all integers a and b,{annZ}{bnnZ}{abnnZ} (d) For all integers a and b,{abmnZ}{annZ}{bnnZ} (e) There exist integers x,y such that 0x25 and 0y25 and 14x29y=75 (f) There exist integers x,y such that 14x+8y=3. (g) For all integers a and b,a24b=2. (h) For all integers a,b,c if a2+b2=c2 then a is even or b is even