Question

The number 55 is multiplied by a 55 digit number that contains only ones. What is the sum of the digits of the product

Answer 1

`Sum = 64`

Explanation:

Given

`n = 55`

`x \to 55\ digit` 1's

Required

The sum of the
`digits` of the
`product`

To do this, we start with smaller numbers

`55*11 = 605` ---- 2 digits

`55*111 = 6105` --- 3 digits

`55*1111 = 61105` -- 4 digits

`55*11111 = 611105` -- 5 digits

`55*111111 = 6111105` --- 6 digits

Notice that as the 1's increase on the left-hand side, the 1's increase on the right-hand side

The difference between the count of both 1's is 2 1's

So, for n digits, we have: 6[n - 2, 1's]05

So, the sum is:

`Sum = 6 + (n - 2) + 0 + 5`

For 55 digit,
`n = 55\\`

So, we have:

`Sum = 6 + (55 - 2) + 0 + 5`

`Sum = 6 + 53 + 0 + 5`

`Sum = 64`