Question

Find all five digit numbers of the form 34X5Y which are divisible by 45.
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Answer 1

it is given that five digits number of the form 34X5Y is divisible by 45.

it means, 34X5Y is also divisible by 9 and 5 too.

we know, divisibility test of 9 : any number is divisible by 9 when sum of all digits included in number is divisible by 9.

so, 34X5Y is divisible by 9 when 3 + 4 + X + 5 + Y = 9n [ where n is positive integer ]

or, 12 + X + Y = 9n .......(1)

from divisibility test of 5 : any number is divisible by 5 when unit digit of number is 0 or 5.

so, 34X5Y is divisible by 5 when Y = 0 or 5

if Y = 0 , from equation (1),

12 + X + 0 = 9n => 12 + X = 9n

for least positive value of X , n = 2

then, X = 9 × 2 - 12 = 6

so, one number will be 34650

if Y = 5 , from equation (1),

12 + X + 5 = 9n => 17 + X = 9n

for least positive value of X, n = 2

then, X = 9 × 2 - 17 = 1

so, other number will be 34155

hence, there are two possible value : 34650 and 34155