Hello everybody!! I had my FTRE(FIITJEE TALENT REWARD EXAM) yesterday.This exam is held all over India and I think you all must have understood the purpose!There was this question that I have no idea how to solve and I was thinking if someone could help me.Anyway,the question is,$\overline{ab}\times \overline{c}=\overline{def}$.These conditions apply on the digits,$(a,b,c,d,e,f):\\ \bullet they\ are\ distinct\\ \bullet none\ of\ them\ is\ equal\ to\ 0\\ \bullet they\ are\ all\ less\ than\ 7.$Find the digits. 5 years, 11 months ago

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$54\times 3=162$

- 5 years, 11 months ago

awesome!Could you please tell how you got that.

- 5 years, 11 months ago

• Firstly since all the digits are distinct so $b, c, f\not = 5$, moreover $d_{max}=3$ ( max occurs at $54\times 6=324$ )so only $a=5$ or $e=5$ .

• $b,c\not = 1$(as that would result in repetition)

• $f\not = 1$ (as 1 would occur only at $u(1\times 1)=1$, $u(7\times 3) =1$, $u(9\times 9)=1$ none of which satisfy the conditions.)

We start with $a=5$,

• If $b=2$ , then $c$ can be $3, 6$

• If $b=3$ , then $c$ can be $2, 4$

• If $b=4$ , then $c$ can be $3$

• If $b=6$ , then $c$ can be $2$

A quick check gives us the solution.

NOTE

• In the last step we cross out some values which give us a bigger unit's digit than 6 or if a repetition occurs.

• $u(x)$ means unit's digit of $x$.

- 5 years, 11 months ago

Thanx dude!

- 5 years, 11 months ago

54 *3 = 162

- 5 years, 11 months ago

yup!that is correct!

- 5 years, 11 months ago

See such questions could by setting few conditions by ourselves-

lets take it that way -- if b =x then c must be what such that f is not equal to c or b

If we put b as 1 then all values of f would be equal to c.

If we put b as 2 we see that c could be = 3

If we put b as 3 we see that c could be = 2 , 4

If we put b as 4 wee see that c could be = 3

5 can never stand on our suppositions

even 6 won't be

so we examine the 4 values of b = 2, 3 , 4 and c also 2 , 3 , 4

then we think for value of a

If a = 1 then it won't be 3 digit number . Similarly even 2 won' t be - Then further we could do trial & error wid the remaining values

hence , 54*3 = 162

- 5 years, 11 months ago

54x3=162

- 5 years, 10 months ago

thanku!

- 5 years, 10 months ago