PLEASE HELP!!

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,ab×c=def\overline{ab}\times \overline{c}=\overline{def}.These conditions apply on the digits,(a,b,c,d,e,f):they are distinctnone of them is equal to 0they are all less than 7.(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.

Note by Adarsh Kumar
4 years, 8 months ago

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54x3=162

Manit Kapoor - 4 years, 8 months ago

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thanku!

Adarsh Kumar - 4 years, 8 months ago

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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

Palash Som - 4 years, 8 months ago

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54 *3 = 162

Palash Som - 4 years, 8 months ago

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yup!that is correct!

Adarsh Kumar - 4 years, 8 months ago

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54×3=16254\times 3=162

Aneesh Kundu - 4 years, 8 months ago

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awesome!Could you please tell how you got that.

Adarsh Kumar - 4 years, 8 months ago

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  • Firstly since all the digits are distinct so b,c,f5b, c, f\not = 5, moreover dmax=3d_{max}=3 ( max occurs at 54×6=32454\times 6=324 )so only a=5a=5 or e=5e=5 .

  • b,c1b,c\not = 1(as that would result in repetition)

  • f1f\not = 1 (as 1 would occur only at u(1×1)=1u(1\times 1)=1, u(7×3)=1u(7\times 3) =1, u(9×9)=1u(9\times 9)=1 none of which satisfy the conditions.)

We start with a=5a=5,

  • If b=2b=2 , then cc can be 3,63, 6

  • If b=3b=3 , then cc can be 2,42, 4

  • If b=4b=4 , then cc can be 33

  • If b=6b=6 , then cc can be 22

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)u(x) means unit's digit of xx.

Aneesh Kundu - 4 years, 8 months ago

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@Aneesh Kundu Thanx dude!

Adarsh Kumar - 4 years, 8 months ago

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