**Quote:**

I tried plugging in 5 and 10 for N and every time got a different remainder. Not sure what am I doing wrong...

Apparently you had a mistake in the calculations.

4(5 + 1)(5 + 8) = 312

312 is 310 + 2. 310 is divisible by 10 and therefore by 5. So the remainder is 2.

4(10 + 1)(10 + 8) = 792

792 is 790 + 2. 790 is divisible by 10 and therefore by 5. So the remainder is 2.

**Here is a way to do the calculations without a calculator**:

4(5 + 1)(5 + 8) = 4 × 6 × 13 = 4 × 78 = 312

4(10 + 1)(10 + 8) = 4 × 11 × 18 = 11 × 72 = 792

**What shortcuts are used here?**1. Multiplication by 11 is quite simple. It is the same as multiply by 10 + the number we multiply.

72 × 11 = 72 × 10 + 72 × 1 = 720 + 72 = 792.

So any other multiplication by 11 will be the same:

77 × 11 = 770 + 77 = 847

273 × 11 = 2730 + 273 = 3003

This trick can also be used for multiplication by 12, 13, 103 etc.

72 × 12 = 720 + 72 × 2 = 720 + 144 = 864

72 × 103 = 7200 + 72 × 3 = 7200 + 216 = 7416

2. Generally, it's easier to multiply by a digit, so instead of

4 × 6 × 13 = 24 × 13

we can do

4 × (6 × 13) = 4 × 78

Note, that calculating 24 × 13 using the previous method is also quite fast here. 24 × 13 = 240 + 24 × 3 = 240 + 72 = 312

3. Multiplication by a digit we do mentally as in long multiplication method:

4 × 78 =

First, we multiply 4 × 8 = 32. So the unit digit will be 2

Then, we multiply 4 × 7 = 28 and add 3 that we got from 32 in the previous step. 28 + 3 = 31

The result is 312.

These are the shortcuts, but in any case you should use the method most convenient to you. And always pick the easiest numbers to plug in first.