Remainders pattern recognition

[user6489]

Hi guys,

I’m having trouble with recognizing the correct pattern in addition with large exponents. For example, this remainders and exponents question.

I got this one wrong because I calculated the unit digit of 3^35 as 3. My reasoning was that 35 is a multiple of 5, and the unit digit of 3^5 is 3, therefore multiples of 3^5 should also have a unit digit of 3. However the solution considers 35 a multiple of 7, thereby calculating the unit digit as 7. Can anyone help me understand why we’d assume 35 is a multiple of 7 and not 5? I’ve made mistakes in similar questions where I miscalculated because I didn’t choose the right base multiple.

[vince]

When finding the units digit of a number, I just find the pattern:

3 = it’s 3

3^2 it’s 9

3^3 it’s 7

3^4 it’s 1

3^5 it’s 3

So the pattern repeats as a group of 4 (3, 9, 7, 1). Then I divide 35 by 4 and see it goes in evenly 8 times, so 4*8 is 32. The 32nd power will therefore end in 1, since that was the end of the 8th repetition of that block of 4. Then I just count my way up to 35… 33 ends in 3, 34 ends in 9, so 35 ends in 7.