Explain why 1.19 is the greatest amount of money it is possible to have without being able to make change for a dollar

Hello,

There are only 2 solutions for $1.19

In cents:

119=1*25+9*10+0*5+4*1

119=3*25+4*10+0*5+4*1

Thus

120=119+1 = (1*25+9*10+0*5+4*1) + 1 = 1*25+9*10+0*5+5*1

but 1*25+7*10+0*5+5*1=100

120=119+1 = (3*25+4*10+0*5+4*1) + 1=3*25+4*10+0*5+5*1

but 3*25+2*10+0*5+5*1=100

So there are no solution for $1.20