HCF • Topic 2 of 2

HCF Applications

Most CAT HCF marks come from recognising the application, not the arithmetic. Three patterns dominate. First, the "largest equal piece" family: the biggest square tile that paves an L×B floor exactly, or the largest container that measures out several volumes — both equal the HCF of the dimensions. Second, the "greatest number dividing a, b, c exactly" is simply HCF(a, b, c). Third, the remainder trick: the greatest number that divides a, b, c leaving the SAME remainder is HCF of the differences, HCF(a−b, b−c, a−c); and the greatest number leaving DIFFERENT remainders r1, r2, r3 is HCF(a−r1, b−r2, c−r3), because subtracting each remainder makes the division exact. A fourth, smaller idea is the HCF of fractions = HCF(numerators)/LCM(denominators), useful when "the largest length that measures these fractional pieces" appears. The exam skill is translation: spot the words "largest", "greatest", "exactly", "leaving remainder" and reach for HCF.

✅ Solved examples

1. Find the largest square tile that can pave a floor 1517 cm by 902 cm exactly, and the number of tiles.
Side = HCF(1517, 902) = 41 cm. Tiles = (1517/41) × (902/41) = 37 × 22 = 814 tiles.
2. Find the greatest number that divides 245 and 1029 leaving remainders 5 and 9 respectively.
Make divisions exact: 245 − 5 = 240, 1029 − 9 = 1020. Answer = HCF(240, 1020). 1020 = 4×240 + 60; 240 = 4×60 + 0 ⇒ HCF = 60.
3. Find the greatest number that divides 70, 147 and 252 leaving the same remainder in each case.
Same remainder ⇒ HCF of differences. 147−70 = 77, 252−147 = 105, 252−70 = 182. HCF(77,105,182) = 7 (77 = 7×11, 105 = 7×15, 182 = 7×26).
4. Find the HCF of the fractions 6/7, 8/9 and 12/21.
HCF = HCF(numerators)/LCM(denominators) = HCF(6,8,12)/LCM(7,9,21) = 2/63.

✏️ Practice — try these, take hints as needed

1. The largest container that can measure 120 L, 180 L and 240 L of three oils an exact number of times has what capacity?
"Measure exactly" ⇒ HCF of the volumes.
HCF(120,180,240).
Pull out 60: HCF(2,3,4)=1.
60 litres
2. Find the greatest number that divides 280 and 1245 leaving remainders 4 and 3 respectively.
Subtract each remainder first.
280−4 = 276, 1245−3 = 1242.
HCF(276, 1242).
138
3. Find the greatest number which divides 43, 91 and 183 leaving the same remainder in each case.
Same remainder ⇒ HCF of differences.
91−43 = 48, 183−91 = 92.
HCF(48, 92).
4
4. A rectangular plot 0.36 km by 0.84 km is to be split into the largest possible equal square plots. Find the side of each square in metres.
Convert to metres: 360 m by 840 m.
Side = HCF(360, 840).
Pull out 120: HCF(3,7)=1.
120 metres
5. Find the HCF of the fractions 9/10, 12/25 and 18/35.
HCF(numerators)/LCM(denominators).
HCF(9,12,18) = 3.
LCM(10,25,35) = 350.
3/350

📝 Topic test — 8 questions

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