Question

Find the sum of the following.

Answer 1

The positive two-digit integers that are not divisible by 3.

Answer 2

Let S = sum of all two digit numbers
and
T = sum of all two digit numbers that divide by 3

Then

S = 10 + 11 + 12 + ... + 99
S = (10 + 99) * 45
S = 4905

while

T = 3 * (4 + 5 + 6 + ... + 33)
T = 3 * [(4 + 33) * 15]
T = 1665

Hence
S – T = 4905 – 1665 = 3240

Answer 3

Or to put it another way:

10 + 11 + 12... + 97 + 98 + 99 = (10 + 99) + (11 + 98) + (12 + 97) ... = 109 + 109 + 109...

You can arrange the two-digit integers that are not divisible by 3 in a similar way:

10 + 11 + 13 ... + 95 + 97 + 98
= (10 + 98) + (11 + 97) + (13 + 95)
= 108 + 108 + 108...

There are 90 two-digit integers, 1/3 of which are divisible by 3. That leaves 30 pairs of integers that sum to 108.

30 * 108 = 3240

Answer 4

why did you multiply by 45?

Answer 5

Because there are 45 pairs.

Answer 6

how did you figure that out?

Answer 7

We have 90 numbers in total (10 to 99) and we're grouping them into pairs of 2.

99-9 =90
90/2=45

Answer 8

this is confusing but thanks for the help.