Question:hard

Harsha received price money in a jackpot. He expressed the price money in the following way.
Consider a number 'N' that can be expressed in the form of prime factors 3, 5, 7 and 11. The total number of factors of N is 32. The difference between the largest and the smallest such number possible, given that each of these prime factors occur at least once, is the price money that Harsha won. What is the price money that Harsha won in the jackpot?

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The 4 exponents+1 must multiply to 32 = \(2^5\); only split into 4 positive parts is 2+1+1+1, so exponents are a permutation of (3,1,1,1). Put the highest power on the biggest prime to maximise, smallest prime to minimise.
Updated On: Jul 20, 2026
  • Rs. 45,640
  • Rs. 91,280
  • Rs. 1,29,360
  • Rs. 1,82,560
  • Rs. 2,73,840
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The Correct Option is C

Solution and Explanation

Factor out the product $3 \times 5 \times 7 \times 11 = 1155$ from both the maximum and minimum values to simplify the arithmetic.

Since the exponents must be a permutation of $(3,1,1,1)$ (as $(a+1)(b+1)(c+1)(d+1)=32=2^5$ forces the exponent set to $\{3,1,1,1\}$), the largest N puts exponent 3 on prime 11:
$N_{max} = 3 \times 5 \times 7 \times 11^3 = 1155 \times 11^2 = 1155 \times 121$

and the smallest N puts exponent 3 on prime 3:
$N_{min} = 3^3 \times 5 \times 7 \times 11 = 1155 \times 3^2 = 1155 \times 9$

So the difference is:
$N_{max} - N_{min} = 1155 \times (121 - 9) = 1155 \times 112$

Computing $1155 \times 112 = 1155 \times 100 + 1155 \times 12 = 1,15,500 + 13,860 = 1,29,360$.

\[\boxed{Price\ money = Rs.\ 1,29,360}\]
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