2 2 2 2 2 2 ⋮ 2 2 ⋯ 2 2 2 2 + 2 2 2 ⋯ 2 2 2 2 ⋯ C B A
The addition shown above representing 2 + 2 2 + 2 2 2 + 2 2 2 2 + ⋯ has 101 rows and the last term consists of 101 number of 2's.
What is A + B + C ?
Problem: courtsey CEMC.
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From first row we get A A =101 2=202 A equaLs 2 So 20 will go to another row B= 💯 2=200 b=0 And the 20 from first which is equal to 220 22 will go to third row Class=99*2=198 And the 22 from second row which gives 220 C=0 Now comes the toughest part Adding them 2plus 0plus0= 2
S ≡ ( 2 + 2 2 + 2 2 2 + ⋯ + # of 2 = 1 0 1 2 2 2 ⋯ 2 2 2 ) (mod 1000) ≡ ( 2 + 2 2 + ( 9 9 ) 2 2 2 ) (mod 1000) ≡ ( 2 + 2 2 + 2 ( 1 0 0 − 1 ) ( 1 0 0 + 1 1 ) ) (mod 1000) ≡ ( 2 + 2 2 + 2 ( 1 0 0 0 0 + 1 1 0 0 − 1 0 0 − 1 1 ) ) (mod 1000) ≡ ( 2 + 2 2 + 2 ( 1 0 0 0 0 + 1 0 0 0 − 1 1 ) ) (mod 1000) ≡ ( 2 + 2 2 + 2 ( − 1 1 ) ) (mod 1000) ≡ ( 2 + 2 2 − 2 2 ) (mod 1000) = 2 (mod 1000)
Thank you.
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2 × 1 0 1 + 2 0 × 1 0 0 + 2 0 0 × 9 9 = 2 2 0 0 2
2 × 1 0 1 + 2 0 × 1 0 0 + 2 0 0 × 9 9 + … + 2 × 1 0 9 9 × 2 + 2 × 1 0 1 0 0 × 1 =
= 2 2 0 0 2 + 1 0 0 0 n = 1 0 0 0 ( n + 2 2 ) + 2 = … 0 0 2 = … A B C
Hence, A = 0, B = 0, C= 2 and
A + B + C = 0 + 0 + 2 = 2