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12 Nov 2024, 03:18
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\(x + y\): 12
\(m + n\): 14
\(x\), \(y\), \(m\), and \(n\) are positive integers. When \(x\) is divided by \(m\), the remainder is 8, and when \(y\) is divided by \(n\), the remainder is 4.
Select the least possible value of x + y for x + y, and select the least possible value of m + n for m + n. Make only two selections, one in each column.
Select the least possible value of x + y for x + y, and select the least possible value of m + n for m + n. Make only two selections, one in each column.
| \(x + y\) | \(m + n\) | |
| 12 | ||
| 13 | ||
| 14 | ||
| 15 | ||
| 16 | ||
| 26 |
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Official Answer
\(x + y\): 12
\(m + n\): 14
12 Nov 2024, 03:19
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Official Solution:
Theory: Positive integer \(a\) divided by positive integer \(d\) yielding a remainder of \(r\) can always be expressed as \(a = qd + r\), where \(q\) is the quotient and \(r\) is the remainder. Note here that \(0 \leq r < d\) (the remainder is a non-negative integer and always less than the divisor).
So, we’d have \(x = mq + 8\), from which it follows that \(x \geq 8\) (the dividend must be greater than or equal to the remainder), and \(m > 8\) (the divisor must be greater than the remainder). The least values of \(x\) and \(m\) are therefore 8 and 9, respectively. For example, \(x = 8\) divided by \(m = 9\) gives the remainder of 8.
Similarly, we’d have \(y = nq + 4\), from which it follows that \(y \geq 4\) (the dividend must be greater than or equal to the remainder), and \(n > 4\) (the divisor must be greater than the remainder). The least values of \(y\) and \(n\) are therefore 4 and 5, respectively. For example, \(y = 4\) divided by \(n = 5\) gives the remainder of 4.
Therefore, the least possible value of \(x + y\) is \(8 + 4 = 12\), and the least possible value of \(m + n\) is \(9 + 5 = 14\).
Correct answer:
\(x + y\) "12"
\(m + n\) "14"
Bunuel
Theory: Positive integer \(a\) divided by positive integer \(d\) yielding a remainder of \(r\) can always be expressed as \(a = qd + r\), where \(q\) is the quotient and \(r\) is the remainder. Note here that \(0 \leq r < d\) (the remainder is a non-negative integer and always less than the divisor).
So, we’d have \(x = mq + 8\), from which it follows that \(x \geq 8\) (the dividend must be greater than or equal to the remainder), and \(m > 8\) (the divisor must be greater than the remainder). The least values of \(x\) and \(m\) are therefore 8 and 9, respectively. For example, \(x = 8\) divided by \(m = 9\) gives the remainder of 8.
Similarly, we’d have \(y = nq + 4\), from which it follows that \(y \geq 4\) (the dividend must be greater than or equal to the remainder), and \(n > 4\) (the divisor must be greater than the remainder). The least values of \(y\) and \(n\) are therefore 4 and 5, respectively. For example, \(y = 4\) divided by \(n = 5\) gives the remainder of 4.
Therefore, the least possible value of \(x + y\) is \(8 + 4 = 12\), and the least possible value of \(m + n\) is \(9 + 5 = 14\).
Correct answer:
\(x + y\) "12"
\(m + n\) "14"
22 Nov 2024, 10:24
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I have revised the question, solution, and formatting by adding more details to enhance clarity.
