Digit Sum

Algebra Level 4

9 × 888888 … 8 ⏟ 2015 8’s × 555555 … 5 ⏟ 2015 5’s \large 9 \times \underbrace{888888\ldots8}_{2015\text{ 8's}} \times \underbrace{555555\ldots5}_{2015 \text{ 5's}}

What is the sum of the digits of the integer above (once multiplied out)?


The answer is 18135.

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Alamuru Ganesh by Alamuru Ganesh , 21, India

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1 solution

Vishal S
Jan 4, 2015

Since

5 ⋅ 8 ⋅ 9 = 360 ⟶ sum of digits = 3 + 6 + 0 = 9 5 \cdot 8 \cdot 9 = 360 \longrightarrow \text{sum of digits } = 3 + 6 + 0 = 9

55 ⋅ 88 ⋅ 9 = 43560 ⟶ sum of digits = 4 + 3 + 5 + 6 + 0 = 18 55 \cdot 88 \cdot 9 = 43560 \longrightarrow \text{sum of digits } = 4+3+5+6+0=18

555 ⋅ 888 ⋅ 9 = 4435560 ⟶ sum of digits = 4 + 4 + 3 + 5 + 5 + 6 + 0 = 27 555 \cdot 888 \cdot 9 = 4435560 \longrightarrow \text{sum of digits } = 4+4+3+5+5+6+0 = 27

5555 ⋅ 8888 ⋅ 9 = 444355560 ⟶ sum of digits = 4 + 4 + 4 + 3 + 5 + 5 + 5 + 6 + 0 5555 \cdot 8888 \cdot 9 = 444355560 \longrightarrow \text{sum of digits } = 4+4+4+3+5+5+5+6+0

We see that the sum of the digits of the resultant of the n n th number in this list is 9 n n . Therefore, the given number's digit sum is 9 ⋅ 2015 = 18135. 9 \cdot 2015 = 18135.

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Proof of your observation ?

Venkata Karthik Bandaru - 5 years, 7 months ago

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We can show that using GP.

Kushagra Sahni - 5 years, 7 months ago

Did the same !!

Akshat Sharda - 5 years, 7 months ago

Same solution...

Dev Sharma - 5 years, 7 months ago

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