find-the-last-four-digits-of-11-15999-

Question Number 157839 by gsk2684 last updated on 28/Oct/21

f i n d t h e l a s t f o u r d i g i t s o f

11^{15999} ?

Answered by Rasheed.Sindhi last updated on 29/Oct/21

11^{15999} \equiv x (m o d 10^{4})

∵ g c d (11, 10^{4}) = 1

∴ 11^{ϕ (10^{4})} \equiv 1 (m o d 10^{4})

ϕ (10^{4}) = 4000

L e a s t n u m b e r n f o r w h i c h

11^{n} \equiv 1 (m o d 10000)

m u s t b e a n y d i v i s o r o f ϕ (10^{4}) = 4000

T r y i n g f o r n = 1, 2, 4, \dots, 500, 800, \dots 4000

w e c a n s e e t h a t

11^{500} \equiv 1 (m o d 10^{4})

∵ 15999 = 500 \times 32 - 1

{(11^{500})}^{32} \equiv {(1)}^{32} (m o d 10^{4})

11^{16000} \equiv 1 (m o d 10^{4})

11^{16000} \equiv 1 + 10^{4} \times 10 (m o d 10^{4})

11^{16000} \equiv 100001 (m o d 10^{4})

D i v i d i n g b y 11

11^{15999} \equiv 9091 (m o d 10^{4})

L a s t 4 d i g i t s o f 11^{5999} a r e : 9091

Commented by gsk2684 last updated on 31/Oct/21

t h a n k s

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