Difference between revisions of "1984 AIME Problems/Problem 2"
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== Solution == | == Solution == | ||
− | 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. |
Any multiple of 5 ends in 0 or 5; since <math>n</math> only contains the digits 0 and 8, the units [[digit]] of <math>n</math> must be 0. | Any multiple of 5 ends in 0 or 5; since <math>n</math> only contains the digits 0 and 8, the units [[digit]] of <math>n</math> must be 0. | ||
− | 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. | + | 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 <math>\frac{8880}{15} = \boxed{592}</math>. | The smallest number which meets these two requirements is 8880. Thus the answer is <math>\frac{8880}{15} = \boxed{592}</math>. |
Revision as of 12:50, 5 March 2011
Problem
The integer is the smallest positive multiple of
such that every digit of
is either
or
. Compute
.
Solution
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 .
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 |