The Biggest Tile That Still Fits Finding Common Ground

Sameeksha wants square tiles to exactly cover a 12ft by 16ft room, with no cutting and no gaps -- the largest such tile size is the Highest Common Factor of 12 and 16, which turns out to be 4ft. Lekhana buys 84kg and 108kg of rice from two farms and wants to repack both into equal-weight bags with nothing left over -- the possible bag weights are exactly the common factors of 84 and 108 (1, 2, 3, 4, 6, 12), and the largest possible bag size is their HCF, 12kg.

Breaking a number down completely into its prime factors reveals every one of its factors directly: 840's prime factorisation lets any subset of those prime factors be multiplied together to check whether a candidate number (like 28) genuinely divides it. A tempting but wrong conjecture -- 'the bigger a number is, the longer its prime factorisation must be' -- is disproven directly by comparing 96 (a longer factorisation, 2x2x2x2x2x3) against 121 (a much shorter one, 11x11), even though 121 is bigger than 96. A number's SIZE and the LENGTH of its prime factorisation are simply unrelated facts about it.

Comparing two numbers' complete prime factorisations directly reveals their HCF: for each prime appearing in BOTH numbers, take whichever power is smaller (the one both numbers can support), and multiply these together. For 112 (2x2x2x2x7) and 84 (2x2x3x7), only 2 and 7 are shared; 112 has four 2's but 84 only has two, so the shared power of 2 is limited to two; multiplying 2x2x7 gives HCF=28. For 96 and 275 (5x5x11), no prime is shared at all, so their HCF is simply 1 -- they are co-prime.

Anshu and Guna cut cloth strips for torans (decorations) of length 6cm and 8cm, and want to know the shortest length that could be measured out exactly using either strip repeatedly -- this is their Lowest Common Multiple, found by taking the LARGEST power of every prime appearing in either number. For 96 (2^5x3) and 360 (2^3x3^2x5), the LCM needs the largest power of 2 seen anywhere (2^5, from 96), the largest power of 3 seen anywhere (3^2, from 360), and the only 5 present (from 360): LCM = 2^5x3^2x5 = 1440.

A ladder-style division method finds HCF and LCM together efficiently: repeatedly divide both numbers by any shared prime factor, continuing until nothing further divides both, with the HCF being the product of all the divisors used, and the LCM extending this by also multiplying in whatever's left over on each side. Guna spots an even faster shortcut for 300 and 150 -- dividing both directly by 50 in one step; Anshu spots a similar shortcut for 630 and 770, dividing by 10 then by 7. Along the way, useful patterns emerge: if one number already divides another exactly (like 6 dividing 18), their HCF is simply the smaller number itself; and the HCF of two numbers always doubles when both numbers are doubled.

Multiplying two numbers' HCF by their LCM always gives back exactly the product of the two original numbers -- checked directly with 105 and 95 (HCF x LCM = 105 x 95), and again with several other pairs, the identity holds every single time for exactly two numbers. This clean relationship does not automatically extend to three numbers at once, and checking it carefully against three-number examples is worth doing explicitly rather than simply assuming a two-number pattern must generalise.