Question

n is a number, such that 2n has 28 factors and 3n has 30 factors. 6n has

• A. 35
• B. 32
• C. 28
• D. None of these

Answer 1

The correct option is A 35

If we visualize the number as $2^p\times 3^q,$ the number of factors would be (p+1)(q+1); For 2n, we realise that $2n=2^{p+1}\times 3^q$ and its number of factors would be (p+2)(q+1) =28. This has multiple possibilities based on the factors of 28. These are: $1\times 28;2\times 14$ and $4\times 7$. Also, $3n=2^p\times 3^{q+1}$ would have (p+1)(q+2)=30 factors. Looking through the factor pairs of 30, we see $1\times 30,2\times 15,3\times 10and5\times 6.$ Considereing both these lists, we can see that if we take p as 5 and q as 3, we get both the conditions fulfilled. Thus, $6n=2^{p+1}\times 3^{q+1}=2^6\times 3^4$ would give us $7\times 5=35$ factors.