2017 AMC 10A Problems/Problem 20
Contents
Problem
Let equal the sum of the digits of positive integer
. For example,
. For a particular positive integer
,
. Which of the following could be the value of
?
Solution 1
Note that . This can be seen from the fact that
. Thus, if
, then
, and thus
. The only answer choice that is
is
.
Solution 2
One divisibility rule that we can use for this problem is that a multiple of will always have its digits sum to a multiple of
. We can find out that the least number of digits the number
is
, with
's and
, assuming the rule above. No matter what arrangement or different digits we use, the divisor rule stays the same. To make the problem simpler, we can just use the
's and
. By randomly mixing the digits up, we are likely to get:
...
...
. By adding
to this number, we get:
...
...
.
Knowing that this number is ONLY divisible by
when
is subtracted, we can subtract
from every available choice, and see if the number is divisible by
afterwards.
After subtracting
from every number, we can conclude that
(originally
) is the only number divisible by
.
So our answer is
.
Solution 3
The number can be viewed as having some unique digits in the front, following by a certain number of nines. We can then evaluate each potential answer choice.
If is correct, then
must be some number
, because when we add one to
we get
. Thus, if
is the correct answer, then the equation
must have an integer solution (i.e.
must be divisible by
). But since it does not,
is not the correct answer.
If is correct, then
must be some number
, because when we add one to
, we get
. Thus, if
is the correct answer, then the equation
must have an integer solution. But since it does not,
is not the correct answer.
Based on what we have done for evaluating the previous two answer choices, we can create an equation we can use to evaluate the final three possibilities. Notice that if , then
must be a number whose initial digits sum to
, and whose other, terminating digits, are all
. Thus, we can evaluate the three final possibilities by seeing if the equation
has an integer solution.
The equation does not have an integer solution for , so
is not correct. However, the equation does have an integer solution for
(
), so
is the answer.
Solution 4 (Intuition)
If adding to
does not carry any of its digits, then
(ex:
. Sum of digits
). But since no answer choice is
, that means
has some amount of
's from right to left.
When , some
's will bump to 0, not affected its
. But the first non-9 digit (from right to left) will be bumped up by 1. So
. For example,
, and the sum of digits
.
Since , that means
. The only answer choice that meets this requirement is
~BakedPotato66
Video Solution
https://youtu.be/zfChnbMGLVQ?t=3996
~ pi_is_3.14
Video Solution
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.