<PRE><B>Question 184700</B>
Let the first number be X and the second number be Y.


We have


X + Y ..........................(1)


X – Y ..........................(2)


X times Y ......................(3)


X / Y ..........................(4)


Since the sum is a whole number, then X has to be a multiple of Y (for X/Y to be an integer).


One of the most obvious choices is if X and Y are equal, so let us try it:


the first would be 2X


the second would be 0


the third would be X^2


the fourth would be 1


Solve X^2 + 2X + 1 = 441


(X + 1)^2 = 441


X + 1 = +/- 21


Therefore X = 20 (or -22), and Y is the same as X, which gives:


X = 20, Y = 20


or


X = -22, Y= -22


20+20; 20-20; 20x20; 20/20 : 40 + 0 + 400 + 1 = 441


-22-22; -22--22; -22x-22; -22/-22 : -44 + 0 + 484 + 1 = 441

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Here is a more in depth solution where X and Y are not assumed equal:


You can do it without supposing they are equal but remember since X/Y is whole, X is a whole multiple (k) of Y so we could say Let X = kY


(1)	would be kY + Y


(2)	would be kY – Y


(3)	would be kYY = kY^2


(4)	would be kY/Y = k


Adding all together would give:


kY^2 + 2kY + k = 441


k(Y^2 + 2Y + 1) = 441


k(Y + 1)^2 = 441


With 441 and (Y + 1)^2 being squared we need k to also be squared and whole.


The factors of 441 which are square are 1, 9, 49 and 441


So by writing the products of 441 as k(Y + 1)^2 using these values for k we have


1 x 21^2 giving k = 1, Y+1 = +/-21; Y = 20 and X = 20 or Y = -22 and X = -22.


9 x 7^2 giving k = 9, Y+1 = +/-7; Y = 6 and X = 54 or Y = -8 and X = -72.


49 x 3^2 giving k = 49, Y+1 = +/-3; Y = 2 and X = 98 or Y = -4 and X = -196.


441 x 1^2 giving k = 441, Y+1 = +/-1, Y = 0 and X = 0 or Y = -2 and X = -882, but we cannot accept 0 as a value for Y since anything divided by 0 is undefined, so only Y = -2, X = -882 is accepted here.


Therefore the solutions are:

X = 	20	;	Y = 	20	;	X + Y =	40	;	X - Y = 	0	;	XY = 	400	;	X/Y = 	1	;	Total = 	441
X = 	-22	;	Y = 	-22	;	X + Y =	-44	;	X - Y = 	0	;	XY = 	484	;	X/Y = 	1	;	Total = 	441
X = 	54	;	Y = 	6	;	X + Y =	60	;	X - Y = 	48	;	XY = 	324	;	X/Y = 	9	;	Total = 	441
X = 	-72	;	Y = 	-8	;	X + Y =	-80	;	X - Y = 	-64	;	XY = 	576	;	X/Y = 	9	;	Total = 	441
X = 	98	;	Y = 	2	;	X + Y =	100	;	X - Y = 	96	;	XY = 	196	;	X/Y = 	49	;	Total = 	441
X = 	-196	;	Y = 	-4	;	X + Y =	-200	;	X - Y = 	-192	;	XY = 	784	;	X/Y = 	49	;	Total = 	441
X = 	-882	;	Y = 	-2	;	X + Y =	-884	;	X - Y = 	-880	;	XY = 	1764	;	X/Y = 	441	;	Total = 	441


So, although k=1 was an obvious guess where X and Y are equal, it simplifies things but guessing 9, 49 and 441 isn’t so obvious.</PRE>