GATE CE 2018 Set 1 — Question 10

Let the common product be $K$. We are given:
$K = A \times B \times C$
$K = B \times G \times E$
$K = D \times E \times F$The letters $A, B, C, D, E, F, G$ are unique integers from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.Consider the prime number 5. If any of the letters is 5, then the product $K$ must be a multiple of 5. This means every one of the three products must contain the factor 5.If $B = 5$, it satisfies conditions 1 and 2. However, for condition 3 to be satisfied, one of $\{D, E, F\}$ must also be 5. Since all letters represent unique integers, this is impossible unless $B = E = 5$, which violates uniqueness.
Similarly, if $E = 5$, it satisfies conditions 2 and 3, but condition 1 would require another 5 among $\{A, B, C\}$.
If any other letter (A, C, G, D, or F) is 5, it only appears in one of the three products, making it impossible for the other products to be multiples of 5 (since 5 is prime and only appears once in the set $\{1, \dots, 9\}$).Therefore, 5 cannot be represented by any of the letters. The same logic applies to the prime number 7. Thus, 5 is the correct choice.