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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.

1 year, 9 months ago

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$54\times 3=162$ · 1 year, 9 months ago

awesome!Could you please tell how you got that. · 1 year, 9 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 $$x$$.

· 1 year, 9 months ago

Thanx dude! · 1 year, 9 months ago

54x3=162 · 1 year, 9 months ago

thanku! · 1 year, 9 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 · 1 year, 9 months ago

54 *3 = 162 · 1 year, 9 months ago

yup!that is correct! · 1 year, 9 months ago