6 Primes

Number Theory Level 3

$a$ , $b$ , $c$ , $d$ , $e$ , $f$ are six distinct prime numbers that satisfies the equation $ab+cd=ef$ .
What is the minimum value of $ef$ ?
(Here is an example set of numbers that satisfies the equation: $a=2,b=53,c=5,d=23,e=13,f=17$ )

The answer is 55.

2 solutions

Giorgos K.
Apr 13, 2018

quick check using Mathematica

p = Prime; z = {};
For[a = 1, a < 2, a++,
For[c = 1, c < 3, c++,
For[e = 1, e < 10, e++,
For[f = 1, f < 10, f++,
For[b = 1, b < 20, b++,
For[d = 1, d < 20, d++,
If[p@a p@b + p@c p@d == p@e*p@f && a != b != c != d != e != f, Print[p@a, " ", p@b, " ", p@c, " ", p@d, " ", p@e, " ", p@f]; Abort[]]]]]]]]

returns 2 17 3 7 5 11

X X
Apr 13, 2018

Let e<f . If e=2 ,then ef>=3x11+5x7=68. f is a prime,so ef is at least 2x37=74.
If e=3 ,then ef>=2x11+5x7=57=3x17,this satisfies the equation.
If e=5 ,then ef>=2x11+3x7=43. f is a prime,so ef is at least 5x11=55.
If e>5 ,then ef>7x11=77.
So,the minimum value of ef is at least 55,and 57 satisfies the equation.
Check 55, 2x17+3x7=55.
Hence,55 is the minimum value of ef.

6 Primes

The answer is 55.

2 solutions

0 pending reports