28 8. The four digit number n = abcd satisfies the following properties: (1) n ≡ 3 (mod 7) (2) n is divisible by 9 (3) the first two digits have the same sum as the last two digits (4) the digit b is smaller than any other digit (5) the digit c is even (a) Use property (1) to explain why 6a + 2b + 3c + d ≡ 3 (mod 7) (2) (b) Use properties (2), (3) and (4) to show that a + b = 9 (4) (c) Deduce that c ≡ 5(a – 1) (mod 7) (2) (d) Hence determine the number n, verifying that it is unique. You must make your reasoning clear. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________