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lys-0071 [83]
2 years ago
9

Let a_n be the integer obtained by writing all the integers from 1 to n from left to right. For example, a_3 = 123 and a_{11} =

1234567891011. Compute the remainder when a_44 is divided by 45.
Mathematics
1 answer:
Artemon [7]2 years ago
3 0

Answer:

  9

Step-by-step explanation:

The 79-digit number of interest can be formulated as a sum of shorter numbers whose remainders can be computed.

<h3>Expanded form</h3>

The expanded form of the number can be written as ...

  a_44 = 01×10^78 +23×10^76 +45×10^74 +67×10^72 +89×10^70 +...

  +10×10^68 +11×10^66 +... +43×10^2 +44×10^0

<h3>Powers of 10</h3>

Each number except the last is multiplied by a power of 10. Powers of 10 modulo 45 are ...

  10 mod 45 = 10

  100 mod 45 = 10

This lets us conclude that any positive power of 10 mod 45 is 10.

<h3>Parts of the sum</h3>

In short, all of the multiplication by powers of 10 can be collapsed to a single multiplication by 10. Hence, the mod 45 value of a_44 will be ...

  a_44 mod 45 = (((01 +23 +45 +67 +89) +10 +11 +12 +... +43)×10 +44) mod 45

  = (((01 +23 +45 +67 +89) mod 45 + sum(10 .. 43) mod 45)×10 +44) mod 45

  = (((225 mod 45) +(901 mod 45))×10 +44) mod 45

  = ((0 +1)×10 +44) mod 45

<h3>Final value</h3>

  a_44 mod 45 = 54 mod 45 = 9

__

<em>Additional comment</em>

The result can be confirmed by a suitable calculator.

__

The sum of the 34 numbers from 10 to 43 is the product of their average value (10+43)/2 = 26.5 and their number, 34. (26.5×34) = 901.

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