The unit digit is 3/4 the hundreds, even number, sum of thousands and tens equal hundreds. ? is 4 digit #?

Question

Full Question is:
What is the 4 digit number that is described in the following clues? The units digit is 3/4 of the hundreds digit. The number is even. The sum of the thousands and the tens digit is equal to the hundreds digit.

Answer 1

Having a four digit number expressed as

ABCD

where A, B, C, and D are integers from 0 to 9. The conditions state that

1. 0.75*B = D
2. D mod 2 = 0
3. A + C = B

In order for condition (2) to be satisfied B/4 must be an even integer. The only values of B which B/4 are integers are 4 and 8, and only 8 yields and even integer. So

B = 8 and D = 6

which satisfy both condition (1) and (2). Condition (3) then becomes

A + C = 8

There is insufficient information to arrive at a unique answer. The various combinations of (A,C) are (0,8), (8,0), (1,7), (7,1), (2,6), (6,2), (3,5), (5,3), (4,4). Reject (0,8) since this would leave a 3 digit number. The following four digit numbers satisfy the conditions above

8806, 1876, 7816, 2866, 6826, 3856, 5836, 4846

Answer 2

Let the number be ABCD.

First clue : D = 3B / 4
Second clue : D is even, that is, it can be represented as 2n, where n can be any digit.

Therefore, 3B / 4 = 2n, or 3B = 8n.

Now, the LHS is divisible by 3, but the RHS can only be divisible by 3 if n is divisible by 3. So we can let n = 3m.

Thus, 3B = 8n = 8 * 3m. So, B = 8m.

For B to be an integer, m must be either 0 or 1, which means B is either 0 or 8. But B can't be equal to zero, because of the third clue, as that would make A = 0 also. Therefore, B = 8.

If B = 8, then D = 3B / 4 = 3 * 8 / 4 = 6.

Third clue : A + C = 8
There's quite a variety here, because A + C could be 1 + 7 or 2 + 6 or 3 + 5 or 4 + 4 or 5 + 3 or 6 + 2 or 7 + 1 or 8 + 0.

So there are 8 answers to ABCD, which are:

1876, 2866, 3856, 4846, 5836, 6826, 7816 or 8806.

Answer 3

a8b6 where a+b=8
8806
7816
6826
5836
4846
3856
2866
1876

take your pick. are you sure there's not something missing?

Answer 4

5836