Is it possible to find?
Find the last three digits of 1 3 7 6 1 3 7 6 .
The answer is 376.
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1 solution
This can be done using CRT too . It becomes bit tedious though...
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Yes CRT is (almost) always tedious to apply in computations. I try to avoid it as a computational tool.. but it is useful conceptually!
Very well explained ,indeed. Sir , while crafting this question I found a peculiar property that any number ending with 376 if raised to any power gives a number which also ends with 376.can you please give some explanation for this property?
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Yes, that is a great observation! Note that 3 7 6 2 ≡ 3 7 6 ( m o d 1 0 0 0 ) since the congruency reduces to 1 ≡ 1 ( m o d 1 2 5 ) and 0 ≡ 0 ( m o d 8 ) . Now we have 3 7 6 n ≡ 3 7 6 ( m o d 1 0 0 0 ) by induction.
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Thank you sir.A very nice explanation.Upvoted!
We have 1 3 7 6 1 3 7 5 ≡ ( 1 1 × 1 2 5 + 1 ) 1 3 7 5 ≡ 1 ( m o d 1 2 5 ) so 1 3 7 6 1 3 7 6 ≡ 1 3 7 6 ≡ 3 7 6 ( m o d 1 0 0 0 ) .