Divisibilty Rule!

Number Theory Level 3

Arish found the value of 3 19 3^{19} to be 11 A 2261467 ‾ \overline{11A2261467} . He found all the digits correctly except the digit denoted by A A . Find A A .


The answer is 6.

Ayush G Rai by Ayush G Rai , 20, India

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2 solutions

Pham Khanh
May 8, 2016

Using the divisibility rules of 9 \large \color{#20A900}{\boxed{9}} , we got that 9 ∣ 3 19 9|3^{19} ⟺ 9 ∣ 11 A 2261467 ‾ \iff 9|\overline{11A2261467} ⟺ 9 ∣ 1 + 1 + A + 2 + 2 + 6 + 1 + 4 + 6 + 7 \iff 9|1+1+A+2+2+6+1+4+6+7 ⟺ 9 ∣ A + 30 \iff 9|A+30 Cause the nearest number that is divisible by 9 9 is 36 so the answer is A = 36 − 30 = 6 \large A=36-30=\LARGE \boxed{6}

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Good solution!!

Ayush G Rai - 5 years, 1 month ago
Rico Lee
May 1, 2016

Thou shalt make an exception and take the easy way out- using a calculator​. Thou has succeeded.

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