What is the sum of all three digit numbers divisible by 8?

The sum of all three-digit numbers that are divisible by 8 is 61,376.

This can be determined without explicitly listing every such number by leveraging mathematical principles related to arithmetic progressions. Here’s a breakdown:

• Identifying the First and Last Numbers:

  • The smallest three-digit number is 100. When divided by 8, the result is 12.5. Therefore, the first three-digit number divisible by 8 is 104 (8 * 13).
  • The largest three-digit number is 999. When divided by 8, the result is 124.875. Hence, the last three-digit number divisible by 8 is 992 (8 * 124).

• Forming an Arithmetic Progression:

  • The three-digit numbers divisible by 8 form an arithmetic progression: 104, 112, 120,..., 992.
  • The common difference between each term is 8.

• Calculating the Number of Terms:

  • To determine the number of terms, we use the formula: number of terms = (last term - first term) / common difference + 1
  • Applying this: (992 - 104) / 8 + 1 = 888 / 8 + 1 = 111 + 1 = 112. There are 112 such numbers.

• Finding the Sum:

  • The sum of an arithmetic progression can be calculated using the formula: Sum = (number of terms / 2) * (first term + last term)
  • Therefore: (112 / 2) (104 + 992) = 56 1096 = 61376

Therefore, based on the reference provided, the calculated sum matches the information.