A teacher used the following activity to convince one of his elementary school students that 1 is not a prime number. Using colored tiles he designated the follow. ing values for the first few primes: a red tile is worth 2; a yellow worth 3; blue worth 5; and green worth 7. He then asked the student to select a few tiles and calculate the product of their values. The student's product was 90 and the teacher, without seeing the tiles, was able to determine that the student had 1 red tile, 2 yellow tiles, and 1 blue. (a) How did the teacher know these were the tiles the student selected? (b) Explain by giving examples why it is not possible to determine the numbers and colors of tiles if 1 is considered to be a prime number and assigned a colored tile.
(a) The teacher knew that the student had selected 1 red tile, 2 yellow tiles, and 1 blue tile because the product of their values, 2 x 3 x 3 x 5, equals 90. Since each yellow tile is worth 3, having 2 yellow tiles gives a product of 3 x 3 = 9. Multiplying this by the value of the blue tile, 5, gives a product of 9 x 5 = 45. The red tile, worth 2, when multiplied by 45, gives a product of 2 x 45 = 90, which matches the product the student calculated. Therefore, the teacher could conclude that the student had selected 1 red tile, 2 yellow tiles, and 1 blue tile.
(b) If 1 is considered to be a prime number and assigned a colored tile, it would not be possible to determine the number and colors of the tiles based solely on the product. This is because the product of the tile values would remain the same regardless of whether 1 is included or not. For example, if 1 is assigned a red tile (worth 2) and we have 1 red tile, 1 yellow tile (worth 3), and 1 blue tile (worth 5), the product would be 2 x 3 x 5 = 30. However, if we remove the red tile (representing 1) and have only 1 yellow tile and 1 blue tile, the product would still be 3 x 5 = 15. Thus, the product alone would not provide enough information to determine the specific numbers and colors of the tiles if 1 is considered prime.