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If m and n are integers and \(\sqrt{mn}= 10\), which of the following cannot be a value of m + n?
A. 25
B. 29
C. 50
D. 52
E. 101
A. 25
B. 29
C. 50
D. 52
E. 101
11 Apr 2022, 11:54
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It is given that m and n are integers, which is very helpful, because it means that we do not have to worry about taking fractional values for m and n such that they satisfy the equation.
Now, the equation itself is \(√mn\) = 10.
Squaring both sides of the equation, we have mn = 100.
As discussed earlier, we only have to consider integer values for m and n such that their product is 100. In other words, we have to look for integral factors of 100.
Now, that’s easily done. Upon prime factorisation, 100 can be written as,
100 = \(2^2 * 5^2\).
The number of factors of 100 = (2+1) * (2+1) = 9.
Listing out the factors of 100, we have: 1, 2, 4, 5, 10, 20, 25, 50, 100.
Remember that we need to find out which of the values CANNOT be the value of m + n.
If m = 1 and n = 100, mn = 100 and m + n = 101. So, answer option 'E' CAN be the value of m + n. Eliminate answer option E.
Similarly, using the factor combinations of (5,20), (4,25) and (2,50), answer options A, B and D can be eliminated.
50 is the only value which cannot be the value of m +n.
The correct answer option is C.
Now, the equation itself is \(√mn\) = 10.
Squaring both sides of the equation, we have mn = 100.
As discussed earlier, we only have to consider integer values for m and n such that their product is 100. In other words, we have to look for integral factors of 100.
Now, that’s easily done. Upon prime factorisation, 100 can be written as,
100 = \(2^2 * 5^2\).
The number of factors of 100 = (2+1) * (2+1) = 9.
Listing out the factors of 100, we have: 1, 2, 4, 5, 10, 20, 25, 50, 100.
Remember that we need to find out which of the values CANNOT be the value of m + n.
If m = 1 and n = 100, mn = 100 and m + n = 101. So, answer option 'E' CAN be the value of m + n. Eliminate answer option E.
Similarly, using the factor combinations of (5,20), (4,25) and (2,50), answer options A, B and D can be eliminated.
50 is the only value which cannot be the value of m +n.
The correct answer option is C.
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