Problems on L.C.M and H.C.F

Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is________?

by
Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is________?

A. 75
B. 81
C. 85
D. 89
Explanation:
Since the numbers are co-prime, they contain only 1 as the common factor.
Also, the given two products have the middle number in common.
So, middle number = H.C.F of 551 and 1073 = 29;
First number = 551/29 = 19
Third number = 1073/29 = 37.
Required sum = 19 + 29 + 37 = 85.

The product of two numbers is 2028 and their H.C.F is 13. The number of such pairs is__________?

by
The product of two numbers is 2028 and their H.C.F is 13. The number of such pairs is__________?

A. 1
B. 2
C. 3
D. 4

Explanation:
Let the numbers be 13a and 13b.
Then, 13a * 13b = 2028 => ab = 12.
Now, co-primes with product 12 are (1, 12) and (3, 4).
So, the required numbers are (13 * 1, 13 * 12) and (13 * 3, 13 * 4).
Clearly, there are 2 such pairs.

The product of two numbers is 4107. If the H.C.F of these numbers is 37, then the greater number is_________?

by
The product of two numbers is 4107. If the H.C.F of these numbers is 37, then the greater number is_________?

A. 101
B. 107
C. 111
D. 185
Explanation:
Let the numbers be 37a and 37b.
Then, 37a * 37 b = 4107 => ab = 3
Now, co-primes with product 3 are (1, 3).
So, the required numbers are (37 * 1, 37 * 3) i.e., (1, 111).
Greater number = 111.

The sum of two numbers is 528 and their H.C.F is 33. The number of pairs of numbers satisfying the above conditions is__________?

by
The sum of two numbers is 528 and their H.C.F is 33. The number of pairs of numbers satisfying the above conditions is__________?

A. 4
B. 6
C. 8
D. 12
Explanation:
Let the required numbers be 33a and 33b.
Then, 33a + 33b = 528 => a + b = 16.
Now, co-primes with sum 16 are (1, 15), (3, 13), (5, 11) and (7, 9).
Required numbers are (33 * 1, 33 * 15), (33 * 3, 33 * 13), (33 * 5, 33 * 11), (33 * 7, 33 * 9).
The number of such pairs is 4.

H.C.F of 3240, 3600 and a third number is 36 and their L.C.M is 24 * 35 * 52 * 72. The third number is___________?

by
H.C.F of 3240, 3600 and a third number is 36 and their L.C.M is 24 * 35 * 52 * 72. The third number is___________?

A. 22 * 35 * 72
B. 22 * 53 * 72
C. 25 * 52 * 72
D. 23 * 35 * 72
Explanation:
3240 = 23 * 34 * 5; 3600 = 24 * 32 * 52
H.C.F = 36 = 22 * 32
Since H.C.F is the product of lowest powers of common factors, so the third number must have (22 * 32 ) as its factor.
Since L.C.M is the product of highest powers of common prime factors, so the third number must have 35 and 72 as its factors.
Third number = 22 * 35 * 72