Question

There are 4 tickets and all these tickets have 14 factors, find the minimum sum of these 4 tickets.

Hint:

To find numbers with exactly 14 factors, try to factorize 14 and use it to form the number’s prime factorization.

Answer 1

Step 1: Understanding the number of factors.
The number of factors of a number \(n\), which has the prime factorization \(n = p_1^{e_1} \times p_2^{e_2} \times \dots \times p_k^{e_k}\), is given by the formula: \[ \text{Number of factors} = (e_1+1)(e_2+1)\dots(e_k+1) \] To find a number with exactly 14 factors, we must factorize 14 in terms of the product of integers that fit the form \((e_1+1)(e_2+1)\dots(e_k+1)\).
Step 2: Finding the factors of 14.
The factorization of 14 is \(14 = 7 \times 2\), so the number must have two prime factors: one raised to the power of 6 and the other to the power of 1. Hence, the smallest such number is \( 2^6 \times 3^1 = 64 \times 3 = 192 \).
Step 3: Finding the minimum sum.
The sum of the first four numbers with exactly 14 factors (such as 192, 576, 864, 1152) gives: \[ 192 + 576 + 864 + 1152 = 1164 \]