Question

The 4-digit number pqrs has the property that pqrs×4=srqp. If p=2 , what is the value of the 3 -digit number qrs ?

Answer 1

We need:
4(2000 + 100q + 10r + s) = 1000s + 100r + 10q + 2 , using only whole numbers
I reduced this to
r = (1333 + 65q - 166s)/10

made up a silly little computer program which found
q = 1
r = 7
s = 8 , and of course we knew p = 2

check: is pqrs×4=srqp

is 4(2000 + 100(1) + 7(10) + 8) = 8000 + 700 + 10 + 2
8712 = 8712 ..... YEA!!!!!

So qrs = 178

Answer 2

Same solution with detailed calculation:

Your numbers are:

1000 p + 100 q + 10 r + s

and

1000 s + 100 r + 10 q + 2

For p = 2 your condition pqrs ∙ 4 = srqp

become

( 1000 ∙ 2 + 100 q + 10 r + s ) ∙ 4 = 1000 s + 100 r + 10 q + 2

( 2000 + 100 q + 10 r + s ) ∙ 4 = 1000 s + 100 r + 10 q + 2

8000 + 400 q + 40 r + 4 s = 1000 s + 100 r + 10 q + 2


8000 must be equal 1000 s

8000 = 1000 s

Divide both sides by 1000

8 = s

s = 8


8000 + 400 q + 40 r + 4 s = 1000 s + 100 r + 10 q + 2

8000 + 400 q + 40 r + 4 ∙ 8 = 1000 ∙ 8 + 100 r + 10 q + 2

8000 + 400 q + 40 r + 32 = 8000 + 100 r + 10 q + 2

Subtract 8000 to both sides

400 q + 40 r + 32 = 100 r + 10 q + 2

Subtract 32 to both sides

400 q + 40 r = 100 r + 10 q - 30

Divide both sides by 10

40 q + 4 r = 10 r + q - 3

Subtract q to both sides

39 q + 4 r = 10 r - 3

Subtract 4 r to both sides

39 q = 6 r - 3

Divide both sides by 3

13 q = 2 r - 1

Divide both sides by 13

q = ( 2 r - 1 ) / 13


2 r - 1 must be divisible with 13

This mean:

2 r - 1 = 13 n

where n is some integer


For n = 1

2 r - 1 = 13 n

2 r - 1 = 13 ∙ 1

2 r - 1 = 13

Add 1 to both sides

2 r = 14

r = 7


For n = 2

2 r - 1 = 13 n

2 r - 1 = 13 ∙ 2

2 r - 1 = 26

Add 1 to both sides

2 r = 27

r = 27 / 2

r = 13.5

It does not satisfy the condition that r must be an integer and 13.5 is not interval 0 to 9.


For n = 3

2 r - 1 = 39

Add 1 to both sides

2 r = 40

r = 20

It not satisfy the condition r = 0 ÷ 9


For n > 1 we get the value of r > 9 so this cannot be the solution because the value for r must be in the interval 0 to 9.

So the only solution that satisfies the condition is r = 0 ÷ 9 is:

r = 7

Put value r = 7 in equation

q = ( 2 r - 1 ) / 13

q = ( 2 ∙ 7 - 1 ) / 13 = ( 14 - 1 ) / 13 = 13 / 13 = 1


Solution:

p = 2 , q = 1 , r = 7 , s = 8

Proof:

pqrs ∙ 4 = srqp

2178 ∙ 4 = 8712