This is a Test of Mathematics Solution Subjective 74 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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The sum of squares of the digits of a three-digit positive number is 146, while the sum of the two digits in the unit’s and the ten’s place is 4 times the digit in the hundred’s place. Further, when the number is written in the reverse order, it is increased by 297. Find the number.

Let is unit’s digit, is ten’s digit and is the hundred’s digit.

Then from the given information we get

… (i)

= …(ii)

= …(iii)

=

From (iii) we get =

=

Now from (ii) we get

=

=

So b is a multiple of 3, b can be 0, 3, 6, 9.

For b = 0

not possible as

b = 0 a = 1 c = 4

then

Similarly for b = 3 a = 2 c = 5

& b = 6 a = 3 c = 6 are not possible.

For b = 9 a = 4 c = 7

So the only solution is 497.