Module 3 · Number System
Chapter 37: Unit Digits
Number System
2 topics
What this chapter covers
The 2 topics below make up Unit Digits on the CAT. Click any topic to open its notes, examples and practice test.
Study material & tests
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Formula Reference Sheet
This chapter
Cyclicity & last digit
| Cycle length 1 (digits 0,1,5,6) | last digit of d^n = d for all n ≥ 1 |
|---|---|
| Cycle length 2 (digits 4,9) | 4: 4,6,4,6… ; 9: 9,1,9,1… |
| Cycle length 4 (digits 2,3,7,8) | 2:2,4,8,6 | 3:3,9,7,1 | 7:7,9,3,1 | 8:8,4,2,6 |
| Reduce exponent (cycle 4) | use n mod 4; if remainder 0, use the 4th term |
| Last digit of a product | multiply only the unit digits, then take its unit digit |
Last two digits power-tools
| Base ends in 1 | (…a1)^n → last two digits = (a×n mod 10) then 1 |
|---|---|
| Base ends in 76 | 76^n always ends in 76 |
| Even base via 24/76 | 2^10 = …24 ; 24^(even) ends 76, 24^(odd) ends 24 |
| Base ends in 25 | 25^n ends in 25 for all n ≥ 1 (odd-power 25 / even patterns) |
| Binomial near 100 | (100±a)^n ≡ (±a)^n + 100·n·(±a)^(n−1) (mod 100) |
CAT reference
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