Difference between revisions of "1976 AHSME Problems/Problem 5"
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− | =Problem | + | == Problem == |
How many integers greater than <math>10</math> and less than <math>100</math>, written in base-<math>10</math> notation, are increased by <math>9</math> when their digits are reversed? | How many integers greater than <math>10</math> and less than <math>100</math>, written in base-<math>10</math> notation, are increased by <math>9</math> when their digits are reversed? | ||
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10</math> | <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10</math> | ||
− | =Solution= | + | == Solution == |
Let our two digit number be <math>\overline{ab}</math>, where <math>a</math> is the tens digit, and <math>b</math> is the ones digit. So, <math>\overline{ab}=10a+b</math>. When we reverse our digits, it becomes <math>10b+a</math>. So, <math>10a+b+9=10b+a\implies a-b=1</math>. So, our numbers are <math>12, 23, 34, 45, 56, 67, 78, 89\Rightarrow \textbf{(C)}</math>.~MathJams | Let our two digit number be <math>\overline{ab}</math>, where <math>a</math> is the tens digit, and <math>b</math> is the ones digit. So, <math>\overline{ab}=10a+b</math>. When we reverse our digits, it becomes <math>10b+a</math>. So, <math>10a+b+9=10b+a\implies a-b=1</math>. So, our numbers are <math>12, 23, 34, 45, 56, 67, 78, 89\Rightarrow \textbf{(C)}</math>.~MathJams | ||
{{AHSME box|year=1976|before=[[1975 AHSME]]|after=[[1977 AHSME]]}} | {{AHSME box|year=1976|before=[[1975 AHSME]]|after=[[1977 AHSME]]}} |
Latest revision as of 13:36, 16 July 2024
Problem
How many integers greater than and less than
, written in base-
notation, are increased by
when their digits are reversed?
Solution
Let our two digit number be , where
is the tens digit, and
is the ones digit. So,
. When we reverse our digits, it becomes
. So,
. So, our numbers are
.~MathJams
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