See attached photo for two number theory problems. One about even numbers and divisibility, the other about modular numbers.

Homework problem Prove the following: Let q, a, b Z be even numbers, q 2 2. We have that q | ab implies q | a or q| b if and only if q = 2 or q = 2p for some prime p> 2. The Universal Product Code (UPC) bar code is made of 12 digits, each of these digits both written below the bar code and encoded by a specific pattern of bars. We call these digits d1, d2, d3,...,d1, d12. The first six digits are a manufacturer identifier, and the next five digits identify a specific product. The last digit is a check digit. To find the check digit, we look at 3d1 + dz + 3dz + d4 + 3dz + de + 3d7 + dg + 3d9 + d10 + 3d11 + d12 = 0 (mod 10). Homework problem Which of the following a valid UPC? Show why the other numbers are not valid UPC's. 0 41196 01012 1 0 52010 00121 2 A bank identification number is a 9 digit number that occurs in the lower left hand corner of bank checks. Let the digits be n1, n2, n3, n4, n5, n6, n7, ng, ng. For a 9 digit number to be a valid bank identification number, it must satisfy 7n1 + 3n2 + 9n3 + 7n4 + 3n5 + 9ng + 7n7 + 3ng +9ng = 0 (mod 10). Homework problem Prove the following: Let q, a, b Z be even numbers, q 2 2. We have that q | ab implies q | a or q| b if and only if q = 2 or q = 2p for some prime p> 2. The Universal Product Code (UPC) bar code is made of 12 digits, each of these digits both written below the bar code and encoded by a specific pattern of bars. We call these digits d1, d2, d3,...,d1, d12. The first six digits are a manufacturer identifier, and the next five digits identify a specific product. The last digit is a check digit. To find the check digit, we look at 3d1 + dz + 3dz + d4 + 3dz + de + 3d7 + dg + 3d9 + d10 + 3d11 + d12 = 0 (mod 10). Homework problem Which of the following a valid UPC? Show why the other numbers are not valid UPC's. 0 41196 01012 1 0 52010 00121 2 A bank identification number is a 9 digit number that occurs in the lower left hand corner of bank checks. Let the digits be n1, n2, n3, n4, n5, n6, n7, ng, ng. For a 9 digit number to be a valid bank identification number, it must satisfy 7n1 + 3n2 + 9n3 + 7n4 + 3n5 + 9ng + 7n7 + 3ng +9ng = 0 (mod 10)