NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 1: Real Numbers - Exercise 1.4 - Long Answer Questions

Question 5

Step 1: Write the numbers.

We are given five numbers: \(n, n+4, n+8, n+12, n+16\).

Step 2: Consider division by 5.

Any integer \(n\), when divided by 5, will leave one of the remainders 0, 1, 2, 3, or 4.

We write this as \(n \equiv r \pmod{5}\), where \(r\) is one of 0, 1, 2, 3, 4.

Step 3: Calculate the remainders of each term.

If \(n \equiv r \pmod{5}\), then:

• \(n \equiv r \pmod{5}\)
• \(n + 4 \equiv r + 4 \pmod{5}\)
• \(n + 8 \equiv r + 8 \equiv r + 3 \pmod{5}\)
• \(n + 12 \equiv r + 12 \equiv r + 2 \pmod{5}\)
• \(n + 16 \equiv r + 16 \equiv r + 1 \pmod{5}\)

Step 4: Observe the pattern.

The five numbers give remainders \(r, r+1, r+2, r+3, r+4\) (all taken modulo 5).

This set covers all possible remainders: 0, 1, 2, 3, 4.

Step 5: Conclusion.

Since exactly one of these remainders is 0, it follows that exactly one of the numbers is divisible by 5.