Solve BOTH of the following problems. Show enough detail and give enough explanation in words, so I can tell how you solved the problem. 1) There are many numbers that divide 109 with a remainder of 4. List all the two-digit numbers that have that property. To divide 109 and have a remainder of 4 you must have the divisor multiplied by the quotient equal to 105, thus resulting in the remainder of 4. To find the two-digit numbers that have that property you need to factor 106 into prime numbers, which is 3 x 5 x7. This will give you all the factors of 105 and have a remainder of 4. The factors are: 3, 5, 7, 15, 21, 35, and 105. The two digit numbers that have this property would be 15, 21, and 35. 2) Thirteen plums weigh as much as two apples and one pear. Four plums and one apple have the same weight as one pear. How many plums have the same weight as one pear? Pl = plums a = apples p = pears 13pl= 2a + p 4 pl + a = p or rewritten a = p-4pl Substitute p=4pl into the first equation for the variable a, thus the first sentence reads 13pl = 2(p-4pl) + p substitution 13pl = 2p ? 8pl + p distributive property of multiplication over subtraction 21pl = 3p add 8pl to both sides of the equation. 2p+p 7pl = p divide by the coefficient, 3 to solve for p, pears Therefore, 7 plums (pl) has the same weight as one pear(p).