Question

4. A number with 50 digits is divisible by 13. 26th digit from left ( same as 25th digit from right ) , is ‘ X ‘ . All other digits ( 49 nos. ) are 1. Find ‘ X ‘ . Explain with proper logic.​

Answer 1

Solution :-

since we know that,

→ 1001 ÷ 13 = Remainder 0 .

so,

→ 1001 * n ÷ 13 = Remainder 0 .

then,

→ 1001 * 111 ÷ 13 = 111111 ÷ 13 = Remainder 0

now,

→ Given 50 digit number = 1111111__ (1x) ___111111 = (24 one's) (1x) (24 one's) .

since,

→ 1111111____ 24 times ÷ 13 = Remainder 0

therefore, we can conclude that, on both sides 24 one's are exactly divisible by 13 .

now,

→ 25th number from left = 1

→ 26th number from left = x

therefore,

→ (1x) ÷ 13 = Remainder 0

only possible value of x is 3 .

hence, 50 digits number 111111___ (13) _____ 11111 will exactly divisible by 13 .

Note :-

→ (6 one's)13 = (111111)13 ÷ 13 = Remainder 0

→ (12 one's)13 ÷ 13 = Remainder 0

→ (18 one's)13 ÷ 13 = Remainder 0

then,

→ (24 one's)13 ÷ 13 = Remainder 0 .

therefore,

→ (24 one's) , 1 (25th) , 3 (26th number)

hence, 26th number from left is equal to 3 .

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