Did anyone say 2?

Algebra Level 3

2 2 2 2 2 2 2 2 2 2 2 2 + 2 2 2 2 2 2 2 C B A \large \begin{array} {r} 2 \\ 2 \ 2 \\ 2 \ 2 \ 2 \\ \vdots \\ 2 \ 2 \ \cdots \ 2 \ 2 \ 2 \ 2 \\ + \ 2 \ 2 \ 2 \ \cdots \ 2 \ 2 \ 2 \ 2 \\ \hline \cdots \ C B A \end{array}

The addition shown above representing 2 + 22 + 222 + 2222 + 2+22+222+2222+\cdots has 101 rows and the last term consists of 101 number of 2's.

What is A + B + C ? A+B+C?

Problem: courtsey CEMC.

8 10 6 2 4

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3 solutions

Zee Ell
Oct 8, 2017

2 × 101 + 20 × 100 + 200 × 99 = 22002 2 × 101 + 20×100 + 200 × 99 = 22002

2 × 101 + 20 × 100 + 200 × 99 + + 2 × 1 0 99 × 2 + 2 × 1 0 100 × 1 = 2 × 101 + 20×100 + 200 × 99 + \ldots + 2 × 10^{99} × 2 + 2 × 10^{100} × 1 =

= 22002 + 1000 n = 1000 ( n + 22 ) + 2 = 002 = A B C = 22002 + 1000n = 1000(n + 22) + 2 = \overline { \ldots 002} = \overline { \ldots ABC}

Hence, A = 0, B = 0, C= 2 and

A + B + C = 0 + 0 + 2 = 2 A + B + C = 0 + 0 + 2 = \boxed {2}

Thank you.

Hana Wehbi - 3 years, 8 months ago
Prajwal Khokale
Oct 16, 2017

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 + 22 + 222 + + 222 222 # of 2 = 101 ) (mod 1000) ( 2 + 22 + ( 99 ) 222 ) (mod 1000) ( 2 + 22 + 2 ( 100 1 ) ( 100 + 11 ) ) (mod 1000) ( 2 + 22 + 2 ( 10000 + 1100 100 11 ) ) (mod 1000) ( 2 + 22 + 2 ( 10000 + 1000 11 ) ) (mod 1000) ( 2 + 22 + 2 ( 11 ) ) (mod 1000) ( 2 + 22 22 ) (mod 1000) = 2 (mod 1000) \begin{aligned} S & \equiv (2+22+222+\cdots+\underbrace{222\cdots222}_{\text{\# of 2}= 101}) \text{ (mod 1000)} \\ & \equiv (2+22+(99)222) \text{ (mod 1000)} \\ & \equiv (2+22+2(100-1)(100+11)) \text{ (mod 1000)} \\ & \equiv (2+22+2(10000+1100-100-11)) \text{ (mod 1000)} \\ & \equiv (2+22+2(10000+1000-11)) \text{ (mod 1000)} \\ & \equiv (2+22+2(-11)) \text{ (mod 1000)} \\ & \equiv (2+22-22) \text{ (mod 1000)} \\ & = \boxed {2} \text{ (mod 1000)} \end{aligned}

Thank you.

Hana Wehbi - 3 years, 8 months ago

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