Final answer:
To prove that 11⁹ –11⁸+11⁷ is divisible by 3 and 37, we can simplify the expression and use the rules of divisibility by 3 and 37.
Step-by-step explanation:
To prove that 11⁹ –11⁸+11⁷ is divisible by 3 and 37, we can simplify the expression to:
11⁹ –11⁸+11⁷ = 1331 – 121 + 11 = 1221
To prove that this expression is divisible by 3, we can use the rule of divisibility by 3, which states that if the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3. In the case of 1221, the sum of the digits is 1 + 2 + 2 + 1 = 6, which is divisible by 3. Therefore, 1221 is divisible by 3.
To prove that 1221 is divisible by 37, we can use long division or check if 37 is a factor of 1221. In this case, 1221 divided by 37 gives a quotient of 33, which is an integer.
Therefore, 1221 is divisible by 37.