In a post-office, stamps of three different denominations of ₹ 7, ₹ 8, ₹ 10…
2014
In a post-office, stamps of three different denominations of ₹ 7, ₹ 8, ₹ 10 are available. The exact amount for which one cannot buy stamps is
Answer: A. 19 — CONCEPTAn amount obtainable with ₹7, ₹8, and ₹10 stamps must have the form 7a + 8b + 10c, where a, b, and c are non-negative integers. To prove that an amount…
- A.
19
- B.
20
- C.
23
- D.
29
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Correct answer: A
CONCEPT
An amount obtainable with ₹7, ₹8, and ₹10 stamps must have the form 7a + 8b + 10c, where a, b, and c are non-negative integers.
To prove that an amount is obtainable, one valid triple (a, b, c) is enough. To prove that it is not obtainable, all feasible values must be checked systematically.
APPLICATION
For ₹19, c can only be 0 or 1 because two ₹10 stamps already total ₹20.
If c = 0, then 7a + 8b = 19. For b = 0, 1, 2, the remainders after subtracting 8b are 19, 11, and 3; none is a non-negative multiple of 7.
If c = 1, then 7a + 8b = 9. For b = 0, 1, the remainders are 9 and 1; neither is a non-negative multiple of 7.
Therefore ₹19 cannot be formed from the available denominations.
CROSS-CHECK
Each other offered amount has a direct construction:
₹20 = ₹10 + ₹10
₹23 = ₹7 + ₹8 + ₹8
₹29 = ₹7 + ₹7 + ₹7 + ₹8
Hence, the exact amount for which stamps cannot be bought is ₹19.