A number has 4 digits. The sum of the first and last digit is twice the second digit. The second digit is 2 less than the third digit. The last digit is twice the first digit. Some of the digits are alike. What is the number?
To find the number, let's break down the given information step by step:
1. The number has 4 digits. Let's represent the digits as ABCD, where A is the first digit, B is the second digit, C is the third digit, and D is the last digit.
2. The sum of the first and last digit is twice the second digit. This can be expressed as: A + D = 2B.
3. The second digit is 2 less than the third digit. This can be expressed as: B = C - 2.
4. The last digit is twice the first digit. This can be expressed as: D = 2A.
Now, let's solve the given equations using the information above:
Substitute equation 3 (B = C - 2) into equation 2:
A + D = 2(C - 2).
Substitute equation 4 (D = 2A) into equation 2:
A + 2A = 2B.
Substitute equation 4 (D = 2A) into equation 1:
A + 2A = 2B.
Combine like terms:
3A = 2B.
Since both equations 2 and equation 3 are equal to 2B, they are equal to each other:
A + 2A = A + D.
Combine like terms:
3A = A + D.
Substitute equation 4 (D = 2A):
3A = A + 2A.
Combine like terms:
3A = 3A.
This equation suggests that A can be any digit.
Let's test out A = 1, A = 2, A = 3, etc., to find the possible values for ABCD.
If A = 1, then D = 2A = 2(1) = 2. However, D should be the last digit, which cannot be 2 according to the given information.
If A = 2, then D = 2A = 2(2) = 4. However, D should be the last digit, which cannot be 4 according to the given information.
If A = 3, then D = 2A = 2(3) = 6. In this case, the last digit is 6, which satisfies the given information.
Now, substitute A = 3 into equation 3:
B = C - 2.
Substitute A = 3 and D = 6 into equation 2:
A + D = 2B.
Simplify the equations:
B = C - 2,
3 + 6 = 2B.
Solve equation 2:
9 = 2B,
B = 4.5.
Since B should be a whole number, there is no valid solution for A = 3.
Thus, there is no valid number that satisfies all the given conditions.