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30 Nov 2021, 06:08
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If \([x]\) is the greatest integer less than or equal to \(x\), and \([√x] = 5\) and \([√y] = 6\), where \(x\) and \(y\) are positive integers, what is the greatest possible value of \(x + y\) ?
A. \(61\)
B. \(81\)
C. \(82\)
D. \(83\)
E. \(85\)
A. \(61\)
B. \(81\)
C. \(82\)
D. \(83\)
E. \(85\)
30 Nov 2021, 06:08
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Official Solution:
If \([x]\) is the greatest integer less than or equal to \(x\), and \([√x] = 5\) and \([√y] = 6\), where \(x\) and \(y\) are positive integers, what is the greatest possible value of \(x + y\) ?
A. \(61\)
B. \(81\)
C. \(82\)
D. \(83\)
E. \(85\)
Some function [] rounds DOWN a number to the nearest integer. For example:
\([2.7] = 2\) because 2 the largest integer less than or equal to 2.7;
\([3] = 3\) because 3 the largest integer less than or equal to 3;
\([-1.7] = -2\) because -2 the largest integer less than or equal to -1.7.
So, \([√x] = 5\) means that 5 is the largest integer less than or equal to \(√x\), which implies that \(5 \leq √x < 6\). For any number from that range, the largest integer less than or equal that number will be 5. Square: \(25 \leq x < 36\). The greatest integer value of \(x\) is 35.
Similarly, \([√y] = 6\) means that 6 is the largest integer less than or equal to \(√y\), which implies that \(6 \leq √y < 7\). For any number from that range, the largest integer less than or equal that number will be 6. Square: \(36 \leq y < 49\). The greatest integer value of \(y\) is 48.
The greatest integer value of \(x+y=35+48=83\)
Answer: D
If \([x]\) is the greatest integer less than or equal to \(x\), and \([√x] = 5\) and \([√y] = 6\), where \(x\) and \(y\) are positive integers, what is the greatest possible value of \(x + y\) ?
A. \(61\)
B. \(81\)
C. \(82\)
D. \(83\)
E. \(85\)
Some function [] rounds DOWN a number to the nearest integer. For example:
\([2.7] = 2\) because 2 the largest integer less than or equal to 2.7;
\([3] = 3\) because 3 the largest integer less than or equal to 3;
\([-1.7] = -2\) because -2 the largest integer less than or equal to -1.7.
So, \([√x] = 5\) means that 5 is the largest integer less than or equal to \(√x\), which implies that \(5 \leq √x < 6\). For any number from that range, the largest integer less than or equal that number will be 5. Square: \(25 \leq x < 36\). The greatest integer value of \(x\) is 35.
Similarly, \([√y] = 6\) means that 6 is the largest integer less than or equal to \(√y\), which implies that \(6 \leq √y < 7\). For any number from that range, the largest integer less than or equal that number will be 6. Square: \(36 \leq y < 49\). The greatest integer value of \(y\) is 48.
The greatest integer value of \(x+y=35+48=83\)
Answer: D
27 Dec 2024, 23:16
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