12

Problems(2)

Divisibility by 11 Generalizes

Source: 2021 AMC 12A Fall #10 / AMC 10A Fall #12

11/11/2021

The base-nine representation of the number

N

27{,}006{,}000{,}052_{\rm nine}

. What is the remainder when

N

is divided by

5?

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>3\qquad<span class='latex-bold'>(E) </span>4

AMCAMC 12AMC10

AMC 10B #12

Source:

11/17/2021

Which of the following conditions is sufficient to guarantee that integers

x

y

, and

z

satisfy the equation

x(x-y)+y(y-z)+z(z-x) = 1?

<span class='latex-bold'>(A)</span>\: x>y

and

y=z

<span class='latex-bold'>(B)</span>\: x=y-1

and

y=z-1

<span class='latex-bold'>(C)</span> \: x=z+1

and

y=x+1

<span class='latex-bold'>(D)</span> \: x=z

and

y-1=x

<span class='latex-bold'>(E)</span> \: x+y+z=1

AMCAMC 10AMC 10 BAMC 12AMC 12 BImportant Announcement