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} =
1 answer:
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
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<em>Additional comment</em>
The result can be confirmed by a suitable calculator.
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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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