Answered

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Let ABC be a 3-digit number such that its digits A, B, and C form an arithmetic sequence. What is the largest integer that divides all numbers of the form ABCABC?

Sagot :

Mlcparra16idn
Let x be the first number in the arithmetic sequence
and y be the difference 

The numbers are:
A,  B  ,  C
x, x+y, x+2y

We add all to check if it is divisible by 3 or 9

[tex]x+x+y+x+2y=3x+3y=3(x+y)[/tex]

Therefore in in the number ABCABC one factor is 3

ABCABC is divisible by 1001 because [tex]ABCABC/1001=ABC[/tex]

Therefore there are two numbers that are always factors of ABCABC which are 3 and 1001, therefore the largest integer that divide all numbers in the form ABCABC is 3*1001 or 3003