Run it (javascript)

function func()
{
	var M=myform.myinput.value;
	var nt=0;
	var outputDiv = document.getElementById("outputDiv");
	outputDiv.innerHTML = "<pre\>";
	if (M<=0 || M>99) {
		alert("Should be >0 and <100");
	} else {
		outputDiv.innerHTML += "<font size=+2 \><b\>M="+M+"</b\></font\><br\><br\>";
		for (var a=1; a<M; a++){
			for (var b=1; b<=a; b++){
				var ab=a+b;
				if (ab>=M) break;
				var k=Math.floor(0.5+(a*b)/ab);
				var diff=Math.abs(a*b-k*ab);
				if (diff==0){
					nt++;
					outputDiv.innerHTML += "a="+a+" b="+b+" k="+k+" nt="+nt+"<br\>";
				}
			}
		}
		outputDiv.innerHTML += "<br\><b\>found "+nt+" solutions</b\>";
	}
	outputDiv.innerHTML += "</pre\>";
}

Here's a javascript code in action. Experiment with it and submit the value for M=1000000000.
[My timing: 42s]