Difference between revisions of "2008 iTest Problems/Problem 4"
Yankeesfan (talk | contribs) (Created page with "== Problem == The difference between two prime numbers is <math>11</math>. Find their sum. == Solution == We know that any prime number, excluding <math>2</math> is congruent ...") |
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== Solution == | == Solution == | ||
| − | We know that any prime number, excluding <math>2</math> is congruent to <math>1 \pmod 2</math> | + | We know that any prime number, excluding <math>2</math>, is congruent to <math>1 \pmod 2</math>. Thus, if both of the primes are not <math>2</math>, their difference would be congruent to <math>0 \pmod 2</math>. Because <math>11 \equiv 1 \pmod 2</math>, one of the primes must be <math>2</math>. It follows that the other prime must then be <math>13</math>. Therefore, the sum of the two is <math>13+2=\boxed{15}</math>. |
== See also == | == See also == | ||
Revision as of 16:15, 7 December 2011
Problem
The difference between two prime numbers is 
. Find their sum.
Solution
We know that any prime number, excluding 
, is congruent to 
. Thus, if both of the primes are not , their difference would be congruent to 
. Because 
, one of the primes must be . It follows that the other prime must then be 
. Therefore, the sum of the two is 
.
