Find the least number which when divided by 2, 5, 9 and 12 leaves a remainder 3 but leaves no remainder when same number is divided by 11.a)281b)357c)360d)363e)224Correct answer is option 'D'. Can you explain this answer?

Question

Answer

(USE THIS METHOD ONLY IF ANY ONE OF THE OPTIONS REMAINDERS AS 0 and don't do this method if all the options got the remainder as 0).
So the question was by dividing 11 there will be no remainder. I divided the numbers in the option with 11 and I got remainders as
a) remainder is 6
b)remainder is 5
c)remainder is 8
d) remainder is 0
e) remainder is different

so the option d leaves a remainder is 0 the answer is 363.
I solved this problem by using this method.

Answer

LCM(2, 5, 9, 12) = 180
So the number will be somewhat = 180x + 3
This number leaves no remainder when divided by 11, so x = 2 to get (180x+3) fully divided by 11
So number = 180*2 + 3

Answer

To find the least number that satisfies the given conditions, we need to find the least common multiple (LCM) of the divisors 2, 5, 9, and 12.

Finding the LCM:
Step 1: Prime factorize each number:
2 = 2^1
5 = 5^1
9 = 3^2
12 = 2^2 * 3^1

Step 2: Take the highest power of each prime factor:
2^2 * 3^2 * 5^1 = 4 * 9 * 5 = 180

So, the LCM of 2, 5, 9, and 12 is 180.

Now, we know that when this number is divided by 2, 5, 9, and 12, it leaves a remainder of 3. But we need to find a number that also leaves no remainder when divided by 11.

Finding a number that leaves no remainder when divided by 11:
Step 1: Subtract the remainder (3) from the LCM (180):
180 - 3 = 177

Step 2: Check if the result is divisible by 11:
177 ÷ 11 = 16 with a remainder of 1

Since the result is not divisible by 11, we need to find the next multiple of 180 that is divisible by 11.

Step 3: Add the LCM (180) to the result (177) until we find a number divisible by 11:
177 + 180 = 357

Step 4: Check if the new result is divisible by 11:
357 ÷ 11 = 32 with a remainder of 5

Since 357 is not divisible by 11, we need to find the next multiple of 180 that is divisible by 11.

Step 5: Add the LCM (180) to the result (357) until we find a number divisible by 11:
357 + 180 = 537

Step 6: Check if the new result is divisible by 11:
537 ÷ 11 = 48 with a remainder of 9

Again, 537 is not divisible by 11, so we repeat the process.

Step 7: Add the LCM (180) to the result (537) until we find a number divisible by 11:
537 + 180 = 717

Step 8: Check if the new result is divisible by 11:
717 ÷ 11 = 65 with a remainder of 2

Finally, we have found a number (717) that satisfies all the given conditions. However, we need to find the least number that satisfies the conditions.

Step 9: Continue adding the LCM (180) until we find the least number divisible by 11:
717 + 180 = 897

Step 10: Check if the new result is divisible by 11:
897 ÷ 11 = 81 with a remainder of 6

Continue the process:

897 + 180 = 1077 (not divisible by 11)
1077 + 180 = 1257 (not divisible by 11)
1257 + 180 = 1437 (not divisible by 11)
1437 + 180 = 1617 (not divisible by 11)
1617 + 180 = 1797

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