Sum of Powers is Perfect Square

Number Theory Level 4

$Q$ , $X$ , $Y$ and $Z$ are $4$ positive integers where:

$Q={ 3 }^{ X }+{ 5 }^{ Y }+{ 7 }^{ Z }$

$Q$ is a perfect square. Find $X+Y+Z$ , and $Q$ is the minimum possible perfect square in the sum above.

The answer is 6.

2 solutions

Rohan Jain
Jul 4, 2014

It is clear that Q is odd, which means its square root would also be odd

lets look at some possible candidates for sqrt(Q)

clearly,
1,2,3,4 are rejected as minimum value of Q is 15

6,8 cannot be the solution (they are even)

5 and 7 would also be impossible (trial and error)

for 9 which means Q=81, put X=3, Y=1, Z=2

P.S. I know this is not a very good method, if anyone has got a different approach please let me know

for x in range(1,10): ... for y in range(1,10): ... for z in range(1,10): ... q =3 x+5 y+7 z ... v = int(q 0.5) ... if v*v==q: print "{0} {1} {2}".format(x,y,z)

Alei Reyes - 6 years, 11 months ago

Using the unit digit logic, we can rule out all possibilities....The possible unit digits come out to be 9,5 and 1. That leaves a simple casework.

Pankaj Joshi - 6 years, 11 months ago

Interesting... I got X=2,Y=2, Z=2 with the same answer.

Ariel Gershon - 6 years, 10 months ago

Sure? Silly mistake. Your solution gives 83, which isn't perfect square.

Sudip Maji - 6 years, 10 months ago

X=0,Y=0,Z=1 thn Q is 9,a perfect square ....bt ans is deffrent

PAPAN PAL - 6 years, 11 months ago

All the integers are positive.0 is not positive.

Adarsh Kumar - 6 years, 11 months ago

Its natural no. 0 is whole no.

Abhishek Dadhich - 6 years, 11 months ago

Bro read qes carefully x y nd z all are non zero

Diwakar Mishra - 6 years, 11 months ago

Josh Speckman
Jul 28, 2014

import math

sat=[]
ints=[]

def frac(n):
  return 1.0*(n-int(n))

for i in range(1, 10):
  for j in range(1, 10):
    for k in range(1, 10):
      x=3**i + 5**j + 7**k
      if frac(math.sqrt(x*1.0))==0:
        sat.append(x)
        ints.append(i+j+k)

print str(ints[sat.index(min(sat))])

Simpler version: for x in range(1,10): for y in range(1,10): for z in range(1,10): q = pow(3,x)+pow(5,y)+pow(7,z) q 1 = math.sqrt(q) if int(q 1)==q_1: print(x+y+z)

David Monk - 6 years, 10 months ago

Sum of Powers is Perfect Square

The answer is 6.

2 solutions

0 pending reports