If Tarun buys only pens costing Rs. 13 each or only pencils costing Rs. 5…

Concept

When the same number leaves remainder r upon division by two coprime divisors a and b, subtracting r makes it divisible by both divisors.

Therefore it is divisible by lcm(a, b), so the original number has the form lcm(a, b) × k + r for some integer k.

Application

1. Let the available amount be N. The conditions give N − 2 divisible by both 13 and 5.

2. Because 13 and 5 are coprime, lcm(13, 5) = 65. Hence every possible amount must have the form N = 65k + 2.

3. For 457, 457 − 2 = 455 = 65 × 7, so it has the required form.

4. For 782, 782 − 2 = 780 = 65 × 12, so it has the required form.

5. For 577, 577 − 2 = 575 = 13 × 44 + 3, so 575 is not divisible by 13 and the required form is absent.

6. For 1042, 1042 − 2 = 1040 = 65 × 16, so it has the required form.

Cross-check

As an independent check, 577 ÷ 13 leaves remainder 5 rather than 2: 577 = 13 × 44 + 5. Thus 577 cannot be the available amount, whereas each other offered value equals 65k + 2.

Result

The amount that cannot be available is 577.