Every letter of the alphabet is given a nonzero numerical value which need not be an integer. The value of a word is the product of the values of its letters. For example, if c=2, a=4, and t=3, then "cat" has the value 2*4*3=24.

Suppose that every pair of homophones has equal values (thus, ate=eight, implying a=igh).

Show that every letter of the alphabet has value 1.