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
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 ?
2 answers
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
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
2 answers