The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder is?

L. C .M of 9, 11, 13 is 1287
On dividing 1294 by 1287, the remainder is 7 .

? 1 must be subtracted from 1294, so that 1293 when divided by 9, 11, 13 leaves in each case the same remainder 6 .

Since 2, 3, 7, 11 are prime numbers and the given expression is
2¹⁰ x 3¹⁰ x 7¹⁷ x 11²⁷

So the number of prime factors in the given expression is (10 + 10 + 17 + 27 ) = 64.

Required length = L. C . M of (64, 80, 96) cm
= 960 cm
= 9.60 m.

Required Number = (LCM of 15, 27, 35 and 42) + 7
= 1890 + 7
= 1897

Required number is divisible by the L . C . M of 12, 18, 21, 28 i.e 252.
Now, greatest number of four digits = 9999
On dividing 9999 by 252, the remainder is 171
? Required number = (9999 - 171) = 9828.

Required number = H.C.F of (1657 - 6) and (2037 - 5)
= H.C.F of 1651 and 2032
= 127

LCM of 5, 10, 15, 20, 25 and 30 is 300. So, the bell will toll together after every 300s (5min).
So, the number of times they toll together = 60/5 + 1=13

LCM of the number is always divisible by the HCF of the same numbers.
So, 78, 104 and 234 are all divisible by 26. Whereas, 144 is not divisible by 26.
Thus, 144 cannot be the LCM of the number whose HCF is 26.

The HCF of m and n is 1, so m and n are prime number.
Let m = 7 and n = 5 ? m + n = 12
HCF of(m + n)and m = HCF of 12 and 7 = 1
Similarly, HCF of (m - n) and n = 1

Suppose two numbers are 3N and 5N.
Then, 3N x 5N = HCF x LCM
? 15N² = 16 x 240
? N² = 256
? N = 16
So, the number are 3N = 3 x 16 = 48 and 5N = 5 x 16 = 80.