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UPSC CDS 1 2021 Math Paper

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Question : 91 of 100

Marks: +1, -0

\text{If the number 413283P759387 is divisible by 13, then what is the value of P?}

3
6
7
8

Solution:

Concept:

Use modular arithmetic or the divisibility rule of 13 to find the missing digit

P

.

Explanation:

The number is

413283P759387

. Write it as

N = 413283 \times 10^7 + P \times 10^6 + 759387

.

First, check

413283

:

13 \times 31791 = 413283

, so

413283 \equiv 0 \pmod{13}

.

Thus

413283 \times 10^7 \equiv 0 \pmod{13}

.

Next, compute

10^6 \pmod{13}

:

10 \equiv 10

,

10^2 \equiv 9

,

10^3 \equiv 12

,

10^4 \equiv 3

,

10^5 \equiv 4

,

10^6 \equiv 1 \pmod{13}

.

Hence

P \times 10^6 \equiv P \pmod{13}

.

Now

759387 \pmod{13}

:

13 \times 58414 = 759382

, remainder

5

, so

759387 \equiv 5 \pmod{13}

.

Therefore,

N \equiv 0 + P + 5 \equiv P + 5 \pmod{13}

.

For

N

to be divisible by

13

, we need

P + 5 \equiv 0 \pmod{13}

, i.e.,

P \equiv 8 \pmod{13}

.

Since

P

is a single digit,

P = 8

.

Answer:

The value of

P

is

8

(Option D).

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