Difference between revisions of "1984 AIME Problems/Problem 2"
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The [[integer]] <math>n</math> is the smallest [[positive]] [[multiple]] of <math>15</math> such that every [[digit]] of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. | The [[integer]] <math>n</math> is the smallest [[positive]] [[multiple]] of <math>15</math> such that every [[digit]] of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. | ||
− | = Solution = | + | == Solution 1== |
Any multiple of 15 is a multiple of 5 and a multiple of 3. | Any multiple of 15 is a multiple of 5 and a multiple of 3. | ||
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− | The sum of the digits of any multiple of 3 must be [[divisible]] by 3. If <math>n</math> has <math>a</math> digits equal to 8, the sum of the digits of <math>n</math> is <math>. For this number to be divisible by 3, < | + | The sum of the digits of any multiple of 3 must be [[divisible]] by 3. If <math>n</math> has <math>a</math> digits equal to 8, the sum of the digits of <math>n</math> is <math>8a</math>. For this number to be divisible by 3, <math>a</math> must be divisible by 3. We also know that <math>a>0</math> since <math>n</math> is positive. Thus <math>n</math> must have at least three copies of the digit 8. |
− | The smallest number which meets these two requirements is 8880. Thus the answer is < | + | The smallest number which meets these two requirements is 8880. Thus the answer is <math>\frac{8880}{15} = \boxed{592}</math>. |
+ | |||
+ | == Solution 2== | ||
+ | |||
+ | Notice how <math>8 \cdot 10^k \equiv 8 \cdot (-5)^k \equiv 5 \pmod{15}</math> for all integers <math>k \geq 2</math>. Since we are restricted to only the digits <math>8,0</math>, because <math>8\equiv -7 \pmod{15}</math> we can't have an <math>8</math> in the optimal smallest number. We can just 'add' fives to quickly get <math>15 \equiv 0 \pmod{15}</math> to get our answer. Thus n is <math>80+800+8000=8880</math> and <math>n/15=8880/15=\boxed{592}</math> | ||
== See also == | == See also == |
Latest revision as of 01:24, 18 July 2024
Contents
Problem
The integer is the smallest positive multiple of
such that every digit of
is either
or
. Compute
.
Solution 1
Any multiple of 15 is a multiple of 5 and a multiple of 3.
Any multiple of 5 ends in 0 or 5; since only contains the digits 0 and 8, the units digit of
must be 0.
The sum of the digits of any multiple of 3 must be divisible by 3. If has
digits equal to 8, the sum of the digits of
is
. For this number to be divisible by 3,
must be divisible by 3. We also know that
since
is positive. Thus
must have at least three copies of the digit 8.
The smallest number which meets these two requirements is 8880. Thus the answer is .
Solution 2
Notice how for all integers
. Since we are restricted to only the digits
, because
we can't have an
in the optimal smallest number. We can just 'add' fives to quickly get
to get our answer. Thus n is
and
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |