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A six-digit number is divisible by 91. If 143 is added to this number, the resulting number is divisible by which of the following number?
13
7
11
91
13
Since the original number is divisible by 91, it is also divisible by the factors of 91 (7 and 13). Because 143 is also divisible by 13 (13 * 11 = 143), the sum must be divisible by 13.
Since the original number is divisible by 91, it is also divisible by the factors of 91 (7 and 13). Because 143 is also divisible by 13 (13 * 11 = 143), the sum must be divisible by 13.
Analyze Divisibility
If a number N is divisible by 91, it is divisible by the prime factors of 91, which are 7 and 13.
Analyze the Added Number
The number added is 143. Check for divisibility of 143: 143 / 13 = 11. Thus, 143 is a multiple of 13.
Apply Property of Divisibility
If N is divisible by 13 and 143 is divisible by 13, then their sum (N + 143) must also be divisible by 13.
B: 7 is a factor of 91, but 143 is not divisible by 7 (143/7 = 20.4), so the sum is not necessarily divisible by 7. C: 11 is a factor of 143, but 91 is not divisible by 11, so the sum is not necessarily divisible by 11. D: 91 is not a divisor of 143, so the sum (N+143) is not necessarily divisible by 91.
A is correct because 13 is a common factor of both the original number (as 91 is a multiple of 13) and the added number 143.
For problems involving divisibility of sums, always break down the added constants into prime factors to find the greatest common divisor shared with the original number.
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