A Factory named color splash is preparing for a large project and has produced…

2025

A Factory named color splash is preparing for a large project and has produced 474 liters of red paint, 426 liters of blue paint and 456 liters of green paint. The factory needs to create the minimum possible number of cans all of equal volume, for each paint color without mixing any of the paints. How many cans will be produced in total if the factory uses all the paints?

  1. A.

    180

  2. B.

    426

  3. C.

    226

  4. D.

    474

Attempted by 42 students.

Show answer & explanation

Correct answer: C

Concept: When several quantities must be split into equal-sized groups using the largest possible group size, so that the total number of groups needed is as small as possible, the group size equals the Greatest Common Divisor (GCD), also called the HCF, of all the quantities. Dividing each quantity by this GCD and adding the resulting quotients gives the minimum total number of groups.

Application: Here the three paint volumes are 474, 426, and 456 liters, and the can volume must be the GCD of all three.

  1. Apply the Euclidean algorithm to 474 and 426: 474 − 426 = 48, so GCD(474, 426) = GCD(426, 48). 426 ÷ 48 = 8 remainder 42, so GCD(426, 48) = GCD(48, 42). 48 ÷ 42 = 1 remainder 6, so GCD(48, 42) = GCD(42, 6). 42 ÷ 6 = 7 remainder 0, so GCD(474, 426) = 6.

  2. Extend to the third volume: since 456 ÷ 6 = 76 remainder 0, the GCD of all three volumes is 6 liters — the largest equal can volume that fits every color exactly.

  3. Divide each color's volume by 6 liters: red = 474 ÷ 6 = 79 cans, blue = 426 ÷ 6 = 71 cans, green = 456 ÷ 6 = 76 cans.

  4. Add the three counts: 79 + 71 + 76 = 226 cans.

Cross-check: Factor each volume into primes: 474 = 2 × 3 × 79, 426 = 2 × 3 × 71, and 456 = 23 × 3 × 19. The only prime factors common to all three are one 2 and one 3, so the GCD is 2 × 3 = 6, confirming the Euclidean-algorithm result. As a second check, the total volume is 474 + 426 + 456 = 1356 liters, and 1356 ÷ 6 = 226, matching the sum of the individual can counts.

Answer: 226 total cans.

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