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The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:
- a)276
- b)299
- c)322
- d)345
Correct answer is option 'C'. Can you explain this answer?
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The H.C.F. of two numbers is 23 and the other two factors of their L.C...
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The H.C.F. of two numbers is 23 and the other two factors of their L.C...
Clearly, the numbers are (23 x 13) and (23 x 14).
Larger number = (23 x 14) = 322.
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The H.C.F. of two numbers is 23 and the other two factors of their L.C...
Given: H.C.F. = 23, L.C.M. factors = 13, 14
To find: Larger of two numbers
Let the two numbers be x and y
Hence, we can write:
x = 23a, y = 23b, where a and b are co-prime
L.C.M. of x and y = 23*13*14
We know that L.C.M. * H.C.F. = x * y
So, we can write:
23*13*14*23 = x * y * 23
x * y = 23^2 * 13 * 14
Now, we have two equations:
x = 23a, y = 23b
x * y = 23^2 * 13 * 14
Substituting the values, we get:
23a * 23b = 23^2 * 13 * 14
a * b = 23 * 13 * 14
Now, since a and b are co-prime, they must be 13 and 14 in some order.
If a = 13 and b = 14, then x = 23a = 23*13 = 299 and y = 23b = 23*14 = 322
If a = 14 and b = 13, then x = 23a = 23*14 = 322 and y = 23b = 23*13 = 299
Hence, the larger of the two numbers is 322.
To find: Larger of two numbers
Let the two numbers be x and y
Hence, we can write:
x = 23a, y = 23b, where a and b are co-prime
L.C.M. of x and y = 23*13*14
We know that L.C.M. * H.C.F. = x * y
So, we can write:
23*13*14*23 = x * y * 23
x * y = 23^2 * 13 * 14
Now, we have two equations:
x = 23a, y = 23b
x * y = 23^2 * 13 * 14
Substituting the values, we get:
23a * 23b = 23^2 * 13 * 14
a * b = 23 * 13 * 14
Now, since a and b are co-prime, they must be 13 and 14 in some order.
If a = 13 and b = 14, then x = 23a = 23*13 = 299 and y = 23b = 23*14 = 322
If a = 14 and b = 13, then x = 23a = 23*14 = 322 and y = 23b = 23*13 = 299
Hence, the larger of the two numbers is 322.
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The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:a)276b)299c)322d)345Correct answer is option 'C'. Can you explain this answer?
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