determine the last 3 digits of the no. 7^999
Answers
Answer:
7≡7(mod10)
72≡9(mod10)
73≡3(mod10)
74≡1(mod10)
75≡7(mod10)
Thus, the last digit of the powers of 7 follows a sequence of period 4 .
{7,9,3,1,7,9,3,1,7,9,3,1,⋯}
How much does 999 differ from a multiple of 4 ?
999=249×4+3
Hence, the last digit of 7999 is the same as the last digit of 73 .
7999≡3(mod10)
Maybe you want the sum of the last two digits? In that case, you have to obtain the congruences with respect to 100 as the modulus.
7≡7(mod100)
72≡49(mod100)
73≡43(mod100)
74≡1(mod100)
75≡7(mod100)
Amazing! The last two digits of the powers of 7 also follow a sequence of period 4 .
{7,49,43,1,7,49,43,1,7,49,43,1,⋯}
The last two digits of 7999 are the same as those of 73 .
7999≡43(mod100)
The sum of the last two digits of 7999 is 7 .
Perhaps you were looking for the last three digits? This makes our task more complicated. Develop a sequence with 1000 as the modulus.
Step-by-step explanation:
7≡7(mod1000)
72≡49(mod1000)
73≡343(mod1000)
74≡401(mod1000)
75≡807(mod1000)
76≡649(mod1000)
77≡543(mod1000)
78≡801(mod1000)
79≡607(mod1000)
710≡249(mod1000)
710≡249(mod1000)
711≡743(mod1000)
712≡201(mod1000)
713≡407(mod1000)
714≡849(mod1000)
715≡943(mod1000)
710≡249(mod1000)
716≡601(mod1000)
717≡207(mod1000)
718≡449(mod1000)
719≡143(mod1000)
720≡1(mod1000)
Finally. This means that the last three digits of the powers of 7 follow a 20 -periodic sequence. I am not going to write that here.
999=49×20+19
The last three digits of 7999 are the same as those of 719 .
7999≡143(mod1000)
The sum would then be 8 .
You can determine any number of digits of the powers of 7 as long as you are willing to put in the effort. There are no shortcuts easier than the use of congruences.
For reference, 7999 is this mammoth number.
7999=
17903666285224547401536497605467631516642643929728373878049928
67887100851881973091892126272205021725188551655993594201095511
84270667518603945415483244077971185735873209516005692320512756
90367373674002100585186524013929659039094717603842178648769418
16548944124064201177382408824402708461241943427461587509767417
35780984509626217985910345077871694079949357039393561781987332
21219196928282686583564440839171715976623187961060015650683284
11307572689853026350721431864080548759856387602032056390488560
84715336603561339170648071516414717293039886163472002338365253
64162333925903115907096918153727348192149342762478373037309722
27562220042754617035588433714671944461188299083896788778820914
47397535224429623316520866911789628001652020897779348090206708
64378594401803116332437523742167528825742898092146260440455132
580195901471547040581591601104468657143