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Mathematics Trick On Number Theory

MATHS TRICK-18

This is useful for those preparing for different Mathematical Entrance exams like CSIR-NET, TIFR, ISI, CMI Entrances. Here it is:


Cycle in the Last Digit
The last digit — the ones place — of a decimal integer d is the remainder of the division d/10. Equivalently, the last digit is the result of the operation mod 10, when following the convention that the least non-negative value — the common residue — is returned. Modular arithmetic, combined with iterative generation of the positive powers of two, allows us to show the cycle in the last digit:

We start with 2, compute it mod 10, multiply that result by 2, compute it mod 10, etc. From this it’s clear the pattern will repeat, once a previous result — 2 in this case, at step 5 — is obtained. This shows that the numbers 2n, n ≥ 1, cycle through the four ending digits 2, 4, 8, and 6.
The cycle implies that powers of two with the same ending digit are related, their exponents differing by a multiple of four:
  • Ends in 2: 21, 25, 29, 213, 217, … .
  • Ends in 4: 22, 26, 210, 214, 218, … .
  • Ends in 8: 23, 27, 211, 215, 219, … .
  • Ends in 6: 24, 28, 212, 216, 220, … .
You can express these relationships more succinctly using the laws of exponents, showing explicitly that the ending digit of the first four positive powers of two determine the ending digit of all positive powers of two:
  • Ends in 2: 21·24k, or 21+4k, k ≥ 0.
  • Ends in 4: 22·24k, or 22+4k, k ≥ 0.
  • Ends in 8: 23·24k, or 23+4k, k ≥ 0.
  • Ends in 6: 24·24k, or 24+4k, k ≥ 0.
Problem : [TIFR-2010] The last digit of  280  is
(a)  2            (b) 4             (c) 6                (d) 8
Solution  :     (c)    Apply this trick here:
Ends in 6: 24·24k,  or  24+4k,  k ≥ 0. Put  k=19. 
  
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