Which is the least number that must be subtracted from 1856, so that the remainder when divided by 7, 12, and 16 is 4.
Show Hint
- 172
- 1361
- 140
- 137
The Correct Option is A
Solution and Explanation
\(\text{LCM}(7, 12, 16) = 336\).
Step 2: Form of the required number:
Let \(x\) be subtracted. Then \(1856 - x = 336k + 4\) for some integer \(k\) (remainder 4 is less than each divisor, so it is valid).
Step 3: Least \(x\):
\(x\) is least when \(1856-x\) is as large as possible. \(1856 - 4 = 1852 = 336\times 5 + 172\), so the largest multiple of 336 not exceeding 1852 is \(1680\). Hence \(1856 - x = 1684\) and \(x = 172\).
Step 4: Check:
\(1684 = 7\times 240 + 4 = 12\times 140 + 4 = 16\times 105 + 4\). The remainder is 4 in each case.
Final Answer:
The correct option is (1): 172.
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