If the number 2² x 5⁴ x 4⁶ x 10⁸ x 6¹⁰ x 15¹² x 8¹⁴ x 20¹⁶ x 10¹⁸ x 25²⁰ is divisible by 10ⁿ, then which one of the following is the maximum value of n?
[amp_mcq option1=”78″ option2=”85″ option3=”89″ option4=”98″ correct=”option4″]
This question was previously asked in
UPSC CAPF – 2018
The given number is N = 2² x 5⁴ x 4⁶ x 10⁸ x 6¹⁰ x 15¹² x 8¹⁴ x 20¹⁶ x 10¹⁸ x 25²⁰.
Let’s express all terms in terms of prime factors 2, 3, 5:
N = 2² * 5⁴ * (2²)⁶ * (2*5)⁸ * (2*3)¹⁰ * (3*5)¹² * (2³)¹⁴ * (2²*5)¹⁶ * (2*5)¹⁸ * (5²)²⁰
N = 2² * 5⁴ * 2¹² * 2⁸ * 5⁸ * 2¹⁰ * 3¹⁰ * 3¹² * 5¹² * 2⁴² * 2³² * 5¹⁶ * 2¹⁸ * 5¹⁸ * 5⁴⁰
Total power of 2: 2 + 12 + 8 + 10 + 42 + 32 + 18 = 124. So, 2¹²⁴.
Total power of 5: 4 + 8 + 12 + 16 + 18 + 40 = 98. So, 5⁹⁸.
Total power of 3: 10 + 12 = 22. So, 3²².
The number is 2¹²⁴ * 3²² * 5⁹⁸.
For this number to be divisible by 10ⁿ = 2ⁿ * 5ⁿ, the maximum possible value of n is min(124, 98) = 98.