Divide the Product
a b and c d are two 2-digit numbers.
It is given that 4 b + a = 1 3 α and 5 d − c = 1 7 β , where α and β are natural numbers.
The largest number that will always divide the product, ( a b x c d ) is:
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2 solutions
4 b + a = 1 3 α ⇒ 1 0 a + b = a b = 1 3 α + ( 9 a − 3 b )
5 d − c = 1 7 β ⇒ 1 0 c + d = c d = 1 7 β + ( 1 1 c − 4 d )
a b × c d = ( 1 3 α + ( 9 a − 3 b ) ) × ( 1 7 β + ( 1 1 c − 4 d ) )
This will be maximum when ( 9 a − 3 b ) = 1 3 × N 1 where N 1 is some natural number and ( 1 1 c − 4 d ) = 1 7 × N 2 where N 2 is some natural number.
Therefore, a b × c d = ( 1 3 × ( α + N 1 ) ) × ( 1 7 × ( β + N 2 ) )
= 1 3 × 1 7 ( α + N 1 ) ( β + N 2 ) = 2 2 1 × N
ab ≡ 1 0 a + b
cd ≡ 1 0 c + d
Therefore,
a b x c d = ( 1 0 a + b ) ( 1 0 c + d )
= ( 1 0 a + ( 4 0 b − 3 9 b ) ) ( 1 0 c + ( 5 1 d − 5 0 d ) )
= ( ( 4 0 b + 1 0 a ) − 3 9 b ) ( 5 1 d − ( 5 0 d − 1 0 c ) )
= ( 1 0 ( 4 b + a ) − 3 9 b ) ( 5 1 d − 1 0 ( 5 d − c ) )
As 4 b + a = 1 3 α and 5 d − c = 1 7 β ;
= ( 1 3 0 α − 3 9 b ) ( 5 1 d − 1 7 0 β )
= ( 1 3 x 1 7 ) ( 1 0 α − 3 b ) ( 3 d − 1 0 β )
= 2 2 1 x ( 1 0 α − 3 9 b ) ( 3 d − 1 0 β )
Hence, the largest number is 221 .