SOURCE

function multiply(num1, num2) {
    if (isNaN(num1) || isNaN(num2)) {
        return '';
    }
    let arr1 = num1.split('');
    let arr2 = num2.split('');

    let res = [];

    for (let i = arr1.length - 1; i >= 0; --i) {
        for (let j = arr2.length - 1; j >= 0; --j) {
            let curIndex = i + j;
            let nextIndex = i + j + 1;
            let sum = arr1[i] * arr2[j] + (res[nextIndex] || 0);

            res[nextIndex] = sum % 10;
            res[curIndex] = Math.floor(sum / 10) +  (res[curIndex] || 0);
        }
    }
    return res.join('').replace(/^0+/, '');
}

console.log(multiply('17', '5'));