Divisible by three" is important because the cell reads a gene in groups of three bases. Each group of three bases corresponds to one of 20 different amino acids used to build a protein. If a mutation disrupts this reading frame, then the entire DNA sequence following the mutation will be read incorrectly.
lactase is the enzyme that leads to the hydrolisis of lactose, when lactose is present the e coli needs to hydrolyse it to simple sugars so it needs to active the gene for lactase synthesis
