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 ?
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
Well, isn't that a puzzling number game! Let's break it down, shall we?
If we know that p=2, that means we have 2qrs × 4 = srqp.
Let's focus on the first digit. The only way for 2qrs × 4 to have srqp as the result is if 2 times 4 gives us a number that starts with an S.
But the only way we can get a number starting with S when multiplying 2 by anything is if the first digit of the result is bigger than 9 (because 2 times 4 is only 8, and the highest we can get is 18).
So, if the first digit is S, that means it must be 1, because we can't have a number that starts with a 2 in this case.
That leads us to 1qrs × 4 = srqp.
Now we need to find the value for qrs, so let's do some math!
When we multiply 1qrs by 4, we know that the first digit of the result must be 1. Therefore, the first digit of qrs must be 2.
We can multiply 124 by 4 and see if it matches the pattern. And indeed, 124 × 4 does equal 496.
So, the 3-digit number qrs is indeed 496. Voilà!
To find the value of the 3-digit number qrs, we can start by understanding the given property of the 4-digit number pqrs.
The property states that pqrs multiplied by 4 is equal to srqp.
We are also given that p = 2.
Let's calculate the value of pqrs.
Since p = 2, the number pqrs can be represented as 2qrs.
Now, let's use the property to set up the equation:
2qrs × 4 = srqp
Expanding the left side of the equation:
8qrs = srqp
We are given that p = 2. Substituting this value:
8qrs = sr2q
Now, let's analyze the equation:
1. qrs is a 3-digit number.
2. Multiplying the 3-digit number qrs by 8, the result should be a 4-digit number sr2q.
From these observations, we can conclude that q must be 1.
Let's substitute this value into the equation:
8(1)rs = sr21
This simplifies to:
8rs = sr21
Analyzing the equation again:
1. rs is a 2-digit number.
2. Multiplying the 2-digit number rs by 8 should result in a 3-digit number sr21.
From these observations, we can deduce that r = 3.
Substituting this value into the equation:
823 = s321
Simplifying further:
8s = s
By comparing the digits on both sides of the equation, we can conclude that s = 8.
Therefore, the 3-digit number qrs is 183.
To find the value of the 3-digit number qrs, we need to solve the given equation pqrs × 4 = srqp.
First, let's substitute p = 2 into the equation. We get 2qrs × 4 = sr2q.
Multiplying out the equation, we have 8qrs = 1000s + 100r + 20q.
Since we know that p = 2, the number pqrs can also be written as 2qrs.
Substituting this back into the equation, we have 8(2qrs) = 1000s + 100r + 20q.
Simplifying further, we get 16qrs = 1000s + 100r + 20q.
Now, since we are looking for the value of the 3-digit number qrs, we can ignore the thousands place (s), as it will not impact the value of qrs.
Hence, we can rewrite the equation as 16qrs = 100r + 20q.
Now let's look for values of q, r, and s that satisfy this equation.
Since q is a 1-digit number, it can only take values from 0 to 9. We can start by substituting each value into the equation and checking if we get a valid result.
Starting with q = 0, we have 16 × 0rs = 100r + 20 × 0.
This simplifies to 0 = 100r, which is not possible since r cannot be zero (leading zeros are not allowed in a 3-digit number).
Next, let's try q = 1. We have 16 × 1rs = 100r + 20 × 1.
This simplifies to 16rs = 100r + 20.
Let's further simplify by dividing both sides of the equation by 4: 4rs = 25r + 5.
We can now go through values of r from 1 to 9 to see if any valid values of s satisfy the equation.
For r = 1, we have 4 × 1s = 25 × 1 + 5, which simplifies to 4s = 30. This is not a valid solution, as s should be a single digit.
Since r = 1 did not yield a valid solution, we can skip r = 1 and move on to r = 2.
For r = 2, we have 4 × 2s = 25 × 2 + 5, which simplifies to 8s = 55.
The only valid solution for s here is s = 7.
Therefore, the value of qrs is 217.
In conclusion, if p = 2, the value of the 3-digit number qrs is 217.
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