9999999999999999o÷99
Answers
Answer:
THE REQUIRED REMAINder is 9,891
[EDITED ONCE]
(i) First of all , we have to be sure of the number of terms in the given product ,which is the same as the number of 9′s in the LAST TERM ,namely 16.
(ii) Next , note that from the 4-th term onwards,every term is congruent to (-1)(mod10,000)
9,999 =(10,000–1) ≡ (-1) (mod 10,000)
99,999 =(100,000–1) ≡ (-1) (mod 10,000)
and so on ...
AND The product of THESE "13" terms ≡ (-1)^13 ≡ (-1) (mod 10,000)
(iii) The product of the FIRST THREE TERMS :
9×99×999
= 999×(99×9)
=1,000×(99×9)-1×(99×9)
= 1.000×{100×9–9}-{100×9–9}
= 1,000×{900–9}-{900–9}
= 891×1,000–891
= 891,000–891
= 890,109
(iv)Hence ,the OVERALL PRODUCT :
≡ -(890,109)(MOD10,000)
≡ -890,109+1,000,000 (MOD 10,000)
≡ 109,891 (mod 10,000)
≡ 109,891–100,00
≡ 9,891 (mod 10,000)
Answer:
The solution is 1.01010