A can X contains 399 litres of petrol and a can Y contains 532 litres of diesel. They are to be bottled in bottles of equal size so that whole of petrol and diesel would be separately bottled. The bottle capacity in terms of litres is an integer. How many different bottle sizes are possible?

To bottle both petrol (399 L) and diesel (532 L) in equal sizes separately, the bottle size must be a common divisor of both numbers.\n\n**Step 1: Find the HCF using the Difference Method**\nFinding factors of 399 and 532 directly can be time-consuming. As seen in the 2020 'sticks' question [17], an elegant shortcut is to take the difference between the numbers.\nDifference = $532 - 399 = 133$.\n\nThe HCF must be a factor of this difference. Let's check if 133 itself divides both numbers:\n- $133 \\times 3 = 399$ (Yes)\n- $133 \\times 4 = 532$ (Yes)\n\nSo, the largest possible bottle is **133 litres**.\n\n**Step 2: Count the possible bottle sizes**\nAny bottle size that divides the HCF (133) will also divide 399 and 532. We just need to find how many factors 133 has.\n\nLet's factorize 133:\n- It ends in 3, so not divisible by 2 or 5.\n- Sum of digits is 7, so not divisible by 3.\n- Try 7: $133 \\div 7 = 19$. \n\nSo, $133 = 7 \\times 19$.\n\nThe factors of 133 are: **1, 7, 19, and 133**.\n\nThere are **4** possible bottle sizes.