If 4p2p34 is divisible by 11 then consider following statement?

Statement I: p is an even digit which is divisible by 2 and 8.

Statement II: p2 + 1 is multiple of 5 and p2 + 2 is the multiple of 11.

Which of the following statements is/are correct?

(a) Only Statement I

(b) Only Statement II

(c) Both Statement I and II

(d) Neither Statement I nor II

Question

If 4p2p34 is divisible by 11 then consider following statement? 

Statement I: p is an even digit which is divisible by 2 and 8. 

Statement II: p2 + 1 is multiple of 5 and p2 + 2 is the multiple of 11.

Which of the following statements is/are correct? 

(a) Only Statement I

(b) Only Statement II

(c) Both Statement I and II

(d) Neither Statement I nor II

ONLiNE UPSC · Accepted Answer

Given number = 4p2p34

To check divisibility by 11, we apply the divisibility rule of 11: the alternating sum of digits (starting from the left) must be divisible by 11. So, the alternating sum is:
4 &ndash; p + 2 &ndash; p + 3 &ndash; 4 = 5 - 2p.
We want 5 &minus; 2p = 0 or multiple of 11

Trying integer values from 0 to 9, only when p = 8, we get:
5 &ndash; 2 &times; 8 = -11, which is divisible by 11. 

So, p = 8.

Now, p = 8 is an even digit and divisible by 2 and 8. 

Statement I is correct.

Also, p2 = 64

So, p2 + 1 = 65 is multiple of 5 and p2 + 2 = 66 which is the multiple of 11. 

Statement II is also correct.

So, option c is correct.

Number Theory

Divisibility Rules

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