Multiple choice

Find the least number which when divided by 18, 21, 30 and 84 leaves 5 as a remainder each time.

  1. 1365

  2. 1265

  3. 1455

  4. 1645
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B Correct answer
Explanation

Find LCM of 18, 21, 30, 84. 18=2*3^2, 21=3*7, 30=2*3*5, 84=2^2*3*7. LCM = 2^2 * 3^2 * 5 * 7 = 4 * 9 * 5 * 7 = 1260. The number is 1260 + 5 = 1265.

AI explanation

To find the least number that leaves the same remainder, we first calculate the Least Common Multiple (LCM) of the divisors 18, 21, 30 and 84. The LCM is calculated by prime factorization: 18 = 2 x 3^2, 21 = 3 x 7, 30 = 2 x 3 x 5, and 84 = 2^2 x 3 x 7, so the LCM is 2^2 x 3^2 x 5 x 7 = 1260. Because the required remainder in each case is 5, we add 5 to this LCM. The required least number is 1265.

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