Discover All Possible 8-Digit Multiples of 2013 with $A = \overline{20abcd13}$

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SUMMARY

The discussion focuses on finding all possible 8-digit multiples of 2013 in the form of $A = \overline{20abcd13}$. The number $A$ must satisfy the condition of being divisible by 2013. The analysis reveals that the digits represented by $a$, $b$, $c$, and $d$ must be chosen such that the entire number remains an 8-digit integer and adheres to the divisibility rule of 2013. The final results yield specific combinations of digits that meet these criteria.

PREREQUISITES
  • Understanding of divisibility rules, particularly for 2013.
  • Familiarity with 8-digit number structures and formatting.
  • Basic knowledge of modular arithmetic.
  • Ability to perform combinatorial calculations for digit selection.
NEXT STEPS
  • Explore the properties of 2013 and its prime factorization.
  • Learn about modular arithmetic and its applications in divisibility.
  • Investigate combinatorial methods for digit arrangement in numbers.
  • Study algorithms for generating and testing multiples of specific integers.
USEFUL FOR

Mathematicians, educators, students studying number theory, and anyone interested in combinatorial mathematics and divisibility problems.

Albert1
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$A=\overline{20abcd13}$ is an 8-digit number ,

also $A$ is a multiple of 2013,please find all possible value of $A$
 
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we know 20132013 multiple of 2013
as 20abcd13 = 20132013 + 201300 * x
so the soultions are 20132013 + 201300 * x as last 2 digits are 13
x= 0 gives 20132013
x =1 gives 20333313
x=2 gives 20534613
x=3 gives 20735913
x=4 gives 20937213
x >=5 gives value outside range
 
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