I can only solve the first. The second, I cannot (yet)---either I'm still not good at Math or the second is posted wrongly.

The first:

1. 10x +y

a. (10x +y)/(xy) = 1 +16/(xy). Mutiply both sides by xy,

10x +y = xy +16 -------------(1)

b. I tried first x is greater than y and I didn't go anywhere, so, let y > x.

(y-x)^2 +xy = 10x +y ----------(2)

Eliminate the xy.

xy from (1) is (10x +y -16), substitute into (2):

(y-x)^2 +10x +y -16 = 10x +y

(y-x)^2 = 16

y-x = 4

y = x+4 ------------(3), substitute in, say, (1):

10x +x+4 = x(x+4) +16

11x +4 = x^2 +4x +16

x^2 -7x +12 = 0

Factoring that,

(x-3)(x-4) = 0

x = 3 or 4

When x = 3:

y = x+4 = 7

So number is 10(3) +7 = 37 -----**

When x = 4:

y = x+4 = 8

So number is 10(4) +8 = 48 -----**

Check 37 against Eq.(2):

(y-x)^2 +xy = 10x +y ----------(2)

(7-3)^2 +(3)(7) =? 10(3) +7

16 +21 =? 30 +7

37 =? 37

Yes, so, OK

Check 48 against Eq.(2):

(y-x)^2 +xy = 10x +y ----------(2)

(8-4)^2 +(4)(8) =? 10(4) +8

16 +32 =? 40 +8

48 =? 48

Yes, so, OK

[Check both 37 and 48 against Eq.(1) if you like.]

Therefore, the number is 37 or 48. ---------------answer.