Question : Let $x$ be the least number divisible by 16, 24, 30, 36, and 45, also $x$ is also a perfect square. What is the remainder when $x$ is divided by 123?
Option 1: 100
Option 2: 40
Option 3: 103
Option 4: 33
Correct Answer: 33
Solution : Given: Let $x$ be the least number divisible by 16, 24, 30, 36, and 45, also $x$ is a perfect square. LCM of 16, 24, 30, 36, and 45 = 24 × 32 × 5 = 720 The number is not a perfect square. ⇒ 24 × 32 × 52 = 720 × 5 = 3600, which is a perfect square. When the number 3600 is divided by 123, the remainder is 33 i.e. $x$ = 33 Hence, the correct answer is 33.
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Question : Let $x$ be the least number which when divided by 8, 9, 12, 14 and 36 leaves a remainder of 4 in each case, but $x$ is divisible by 11. The sum of the digits of $x$ is
Option 1: 5
Option 2: 6
Option 3: 9
Option 4: 4
Question : What is the sum of the digits of the least number which when divided by 15,18, and 36 leaves the same remainder 9 in each case and is divisible by 11?
Option 1: 15
Option 2: 16
Option 3: 18
Option 4: 17
Question : What is the least number which when divided by 15, 18 and 36 leaves the same remainder 9 in each case and is divisible by 11?
Option 1: 1269
Option 2: 1071
Option 3: 1089
Option 4: 1080
Question : The least number, which when divided by 5, 6, 7 and 8 leaves a remainder of 3, but when divided by 9, leaves no remainder, is:
Option 1: 1677
Option 2: 1683
Option 3: 2523
Option 4: 3363
Question : Let $x$ be the least 4-digit number which when divided by 2, 3, 4, 5, 6 and 7 leaves a remainder of 1 in each case. If $x$ lies between 2800 and 3000, then what is the sum of the digits of $x$?
Option 1: 12
Option 2: 15
Option 3: 13
Option 4: 16
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