**Explanation :**The prime numbers common to given numbers are 2,5 and 7. H.C.F. = 22 x 5 x72 = 980.

**Explanation :**108 = \(2^2\) x \(3^3\), 288 = \(2^5\) x \(3^2\) and 360 = \(2^3\) x 5 x\(3^2\). H.C.F. = \(2^2\) x \(3^2\) = 36.

**Explanation :**

**Explanation :**H.C.F. of 391 and 667 is 23. On dividing the numerator and denominator by 23,

we get :

**Explanation :**L.C.M. = Product of highest powers of 2, 3, 5, 7 and 11 = \(2^3\) x \(3^3\) x \(5^3\) x \(7^4\) x 11

**Explanation :**72 = \(2^3\) x \(3^2\),

108 = \(3^3\) x\( 2^2\),

2100 = \(2^2\) x \(5^2\) x 3 x 7.

L.C.M. = \(2^3\) x \(3^3\) x\( 5^2\) x 7 = 37800.

**Explanation :**

**Explanation :**

**Explanation :**Making the same number of decimal places, the given numbers are 0.63, 1.05 and 2.10. Without decimal places, these numbers are 63, 105 and 210.

L.C.M. of 63, 105 and 210 is 630.

L.C.M. of 0.63, 1.05 and 2.1 is 6.30.

**Explanation :**Let the required numbers be 15.x and llx. Then, their H.C.F. is x. So, x = 13. The numbers are (15 x 13 and 11 x 13) i.e., 195 and 143.

**Explanation :**Other number = \(\frac{11 * 693}{77}\) = 99

**Explanation :**Required length = H.C.F. of 495 cm, 900 cm and 1665 cm.

495 = \(3^2\) x 5 x 11,

900 = \(2^2\) x \(3^2\) x \(5^2\),

1665 = \(3^2\) x 5 x 37.

H.C.F. = 32 x 5 = 45.

Hence, required length = 45 cm.

**Explanation :**Required number = H.C.F. of (1657 - 6) and (2037 - 5) = H.C.F. of 1651 and 2032

**Explanation :**Required number = H.C.F. of (132 - 62), (237 - 132) and (237 - 62) = H.C.F. of 70, 105 and 175 = 35.

**Explanation :**

**Explanation :**Required number = (L.C.M OF 6,7,8,9,12) + 1

L.C.M = 3 X 2 X 2 X 7 X 2 X 3 = 504

. Hence required number = (504 +1) = 505.

**Explanation :**The Largest number of four digits is 9999.

Required number must be divisible by L.C.M. of 12,15,18,27 i.e. 540. On dividing 9999 by 540,we get 279 as remainder .

Required number = (9999-279) = 9720.

**Explanation :**Smallest number of five digits is 10000.

Required number must be divisible by L.C.M. of 16,24,36,54 i.e 432,

On dividing 10000 by 432,we get 64 as remainder.

Required number = 10000 +( 432 – 64 ) = 10368.

**Explanation :**Here,(20-14) = 6,(25 – 19)=6,(35-29)=6 and (40-34)=6.

Required number = (L.C.M. of 20,25,35,40) – 6 =1394.

**Explanation :**L.C.M. of 5,6,7,8 = 840.

Required number is of the form 840k + 3

Least value of k for which (840k + 3) is divisible by 9 is k = 2.

Required number = (840 X 2 + 3)=1683

**Explanation :**Interval of change = (L.C.M of 48,72,108)sec.=432sec.

So, the lights will agin change simultaneously after every 432 seconds i.e,7 min.12sec

Hence , next simultaneous change will take place at 8:27:12 hrs.

**Explanation :**