GATE2018 CE-1: GA-10

Question

Each of the letters arranged as below represents a unique integer from $1$ to $9.$ The letters are positioned in the figure such that $(A \times B \times C), (B \times G \times E)$ and $(D \times E \times F)$ are equal. Which integer among the following choices cannot be represented by the letters $A, B, C, D, E, F  \ or \ G?$

A		D
B	G	E
C		F

• A. $4$
• B. $5$
• C. $6$
• D. $9$

Comments

ans is 5?

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B) 5 should be the correct answer.

As if you put G = 5, then you will never be able to fill such a number in either coloums which have 5 in its factor. And similar cas with 7. Because both are prime no.

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Out of this numbers 5 and 7 cannot be selected, because they are the prime numbers whose has no multiples other than itself present in the sequence of 1-9.

From the options '5' could be concluded directly.

A*B*C=B*G*E

This implies A*C=G*E , if you assumes A=5 then A*C has its ones place as either 0 or 5.

G and E  has to selected in such a way that G * E ones place is 0 or 5 ,there is no other pair in 1-9 (other than 5) that would generate a ones place as 0 or 5. This can be applied to all Variables, if '5' is present ,above equality cannot be satisfied.

That is the reasoning I followed

Answer 1

from entire numbers 1 to 9, only 5,7 are not used.

Answer 2

Here it is given that,

ABC=BGE=DEF

this means that,

AC=GE  and BG=DF

Now 4*2=1*8

Here,i can assign A=4,C=2,G=1,E=8

Now,6*3=2*9

Here,i can assign A=6,C=3,G=2,E=9

In above 2 cases,i can find a assignment of a letter to 4,6,9. Only number left is 5. Hence it is the answer.

Also if A or C is 5, then the product GE must be one of 5,10,15,20,25,30,35 40,45 which requires one of G or E to be 5. This violates,unique assignment property given in question.