A cuboid of dimensions 50 cm, 150 cm, 175 cm can be divided into how many identical largest cubes?

Correct — A. Identical largest cubes means the edge must divide all three dimensions exactly and be as large as possible — that is the HCF.

50 = 2 × 5²
150 = 2 × 3 × 5²
175 = 5² × 7
Common to all three: 5² = 25 cm

Cubes along each edge: 50 ÷ 25 = 2, 150 ÷ 25 = 6, 175 ÷ 25 = 7

Number of cubes = 2 × 6 × 7 = 84 → option (a)

Why the others are wrong

• (b)
  75 — 75 cannot come from any cube edge. Each piece would have to hold 1,312,500 ÷ 75 = 17,500 cm³, and 17,500 is not a perfect cube, so no edge length cuts the solid into 75 equal cubes.
• (c)
  90 — 90 pieces would need 14,583.3 cm³ each (1,312,500 ÷ 90), which is not even a whole number of cubic centimetres, let alone a cube.
• (d)
  85 — 85 factors as 5 × 17, and 17 divides none of 50, 150 or 175. The count is a product of the pieces along the three edges, so a prime factor that fits no edge cannot appear in it.

"Largest identical cubes" pins the edge down completely: it must divide 50, 150 and 175 with nothing left over, and be the biggest such length. That is the definition of the highest common factor.

With the edge at 25 cm, the counting is three divisions multiplied together, one per dimension — 2 × 6 × 7.

The volume route gives the same number: 50 × 150 × 175 = 1,312,500 cm³ divided by 25³ = 15,625 cm³ is 84. It is a good check and a slow first move.

Key facts

• The edge of the largest identical cube cut from a cuboid is the HCF of its three dimensions.
• HCF(50, 150, 175) = 25.
• The cuboid takes 2, 6 and 7 cubes along its three edges.
• Total volume 1,312,500 cm³ divided by cube volume 15,625 cm³ gives 84.

Common traps

• Using the LCM instead of the HCF, which asks the opposite question.
• Taking the HCF of two dimensions and forgetting that 175 has no factor 2 or 3.
• Adding 2 + 6 + 7 instead of multiplying them.

No directly related past PYQ was found.