9 8. Recall that tn denotes the nth triangular number. Prove the following identity:tn+2k = tn (mod k)9. Determine the remainder when 3701 is divided by 86.10. The Universal Product Code (UPC), used on nearly all retail packaging, consists of a string of 11 digitsa1aza3...a11 followed by a twelfth check digit, a12, that is calculated in the following manner:a12 = 3a1 + a2 + 3a3 + a4 + 3a5 + a6 + 3a7 + ag + 3ag + a10 + 3a11 (mod 10)Calculate the check digit for the following UPC:85911 325908.Find the missing digit from the following UPC:32431 88146511. Billy the Kid and 16 additional outlaws robbed a train carrying gold bars out West. They attempt todivide the gold bars evenly, but three gold bars were left over. Unable to decide how to distribute thethree extra gold bars, a fight broke out and one outlaw was killed. The outlaws then tried again to dividethe bars evenly, but 15 were left over. Once again, a fight broke out concerning the extra gold barsleaving another outlaw dead. Finally, the remaining outlaws were then able to distribute the gold barsevenly. What is the least possible amount of gold bars they could have stolen?-12. (5 points (bonus)) An old receipt shows that 72 holiday turkeys were purchased for $2x79.1y where thedigits x and y are illegible. Use congruence theory to find the unit price paid per turkey.T UG E

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