Question:medium

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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32 as a product of four factors, each at least 2, splits only as 2, 2, 2 and 4. So one prime gets exponent 3 and the rest get exponent 1; put exponent 3 on the largest prime for the biggest N, and on the smallest prime for the smallest N.
Updated On: Jul 21, 2026
  • Rs. 45,640
  • Rs. 91,280
  • Rs. 1,29,360
  • Rs. 1,82,560
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The Correct Option is C

Solution and Explanation

N is made only from the primes 3, 5, 7 and 11, each appearing at least once, and its total number of factors is 32. Writing \(N = 3^a5^b7^c11^d\), the factor count formula \((a+1)(b+1)(c+1)(d+1) = 32\) has to hold, with a, b, c, d each at least 1.

Since 32 breaks down as \(2^5\), and we need four numbers, each at least 2, multiplying to give 32, let's list out the only way this split can happen:

  1. 2, 2, 2, 4: product = 2 x 2 x 2 x 4 = 32. This is the only combination of four factors, each at least 2, that multiplies to 32, since any other split, like 2, 2, 8, 1 or 16, 2, 1, 1, needs a factor below 2, which is not allowed here.

So exactly one of a, b, c, d equals 3, since 4 - 1 = 3, and the rest equal 1, since 2 - 1 = 1. This exponent of 3 can be handed to any one of the four primes, giving four possible values of N. Since we want the biggest and smallest of these four, we just need to try putting the exponent 3 on the biggest prime, 11, and on the smallest prime, 3:

Prime getting exponent 3Resulting N
3\(3^3 \times 5 \times 7 \times 11 = 27 \times 385 = 10,395\)
11\(3 \times 5 \times 7 \times 11^3 = 105 \times 1331 = 1,39,755\)

Giving the exponent 3 to a bigger prime always produces a bigger N, and giving it to the smallest prime always produces the smallest N, so the other two combinations, exponent 3 on 5 or on 7, do not even need checking; they land strictly between these two values.

Let's summarize:

  • 32 splits into four factors of at least 2 in only one way: 2, 2, 2 and 4.
  • This means one prime gets exponent 3 and the rest get exponent 1.
  • Putting exponent 3 on the largest prime, 11, gives the largest N, and on the smallest prime, 3, gives the smallest N.

The difference is 1,39,755 - 10,395 = Rs. 1,29,360, which is the price money Harsha won.

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