find the unit digit 1⁵⁵+2⁵⁵+3⁵⁵+4⁵⁵+-----------+99⁵⁵
Answers
Given :- find the unit digit 1⁵⁵+2⁵⁵+3⁵⁵+4⁵⁵+-----------+99⁵⁵ ?
Answer :-
we know that,
→ 1² = 1 = unit digit 1 .
→ 2² = 4 = unit digit 4.
→ 3² = 9 = unit digit 9.
→ 4² = 16 = unit digit 6.
→ 5² = 25 = unit digit 5.
→ 6² = 36 = unit digit 6 .
→ 7² = 49 = unit digit 9.
→ 8² = 64 = unit digit 4.
→ 9² = 81 = unit digit 1.
→ 10² = 100 = unit digit 0.
dividing power by 4 we get,
- → 55 ÷ 4 = 3 remainder .
so, 1⁵⁵+2⁵⁵+3⁵⁵+4⁵⁵+-----------+99⁵⁵ can be written as ,
→ 1³ + 2³ + 3³ + 4³ + _______ + 99³
→ taking unit digits now,
→ (1 + 8 + 7 + 4 + 5 + 6 + 3 + 2 + 9) 10 times ,
→ 45 + 45 + 45 ___ 10 times
→ 45 * 10
→ 450
→ 0 in last.
therefore, unit digit will be zero.
Learn more :-
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Given : 1⁵⁵+2⁵⁵+3⁵⁵+4⁵⁵+-----------+99⁵⁵
To Find : unit digit
Solution:
1⁵⁵ = 1
2⁵⁵ = ((2)⁴)¹³ * 2³ = 6¹³ *8 = 6* 8 = 48 ( 2⁴ = 16 and 6 raised to power any number 6 )
3⁵⁵ = ((3)⁴)¹³ * 3³ = 1¹³ *27 = 1* 27 = 27 ( 3⁴ = 81 )
4⁵⁵ = ((4)⁴)¹³ * 4³ = 6¹³ *64 = 6* 4 = 24 ( 3⁴ = 81 )
5⁵⁵ = 5
6⁵⁵ = 6
7⁵⁵ = ((7)⁴)¹³ * 7³ = 1¹³ *243 = 243 ( 7⁴ = 2401 )
8⁵⁵ = ((8)⁴)¹³ * 8³ = 6¹³ * 512 = 6* 2 = 12
9⁵⁵ = ((9)⁴)¹³ * 9³ = 1¹³ * 729 = 9
0 raised to power any number = 0
Unit Digits are 1 , 8 , 7 , 4 , 5 , 6 , 3 , 2 , 9
Adding all = 45
numbers ending with 1 to 9 are 10 each
Hence 45 * 10 = 450 so unit digit is 0
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