Three Primes
Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?
Age
14 to 16
Challenge level
How many sets of three prime numbers have the property that the product of the three numbers is exactly five times their sum? (The order of the three numbers is not important).
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Answer: 2, 5 and 7
Searching for possibilities
Product is $5\times$sum, so product is a multiple of $5$
$\therefore$ one of the numbers is $5$
$5\times\underline{ }\times\underline{ }=5\times(5+\underline{ }+\underline{ })$
$\Rightarrow \underline{ }\times\underline{ }=5+\underline{ }+\underline{ }$
$3\times7=21$ $5+3+7=15$ numbers too big
$3\times5=15$ $5+3+5=13$ numbers too small
$5\times5=25$ $5+5+5=15$ numbers too big
$2\times7=14$ $5+2+7=14$ yes
All reasonable possibilities tested
Using algebra
Let $p$, $q$ and $r$ be three prime numbers such that $pqr=5(p+q+r)$. Then one of the prime numbers must be $5$, say $r$.
This implies that $5pq=5(p+q+5)\Rightarrow pq=p+q+5\Rightarrow pq-p-q+1=6\Rightarrow (p-1)(q-1)=6$.
Therefore either $p-1=1$ and so $q-1=6$ i.e. $(p,q)=(2,7)$ (or vice versa) or $p-1=2$ and so $q-1=3$ i.e. $(p,q)=(3,4)$ (or vice versa). But $4$ is not prime, so the only triple of primes which satisfies the condition is $(2,5,7)$.