Step-by-step explanation:
remember the meaning of an exponent : the basic term is multiplied by itself that many times.
e.g.
5⁶ = 5×5×5×5×5×5
so,
2^a or 2^p is a multiplication of only 2s.
5^b or 5^q is a multiplication of only 5s.
7^c or 7^r is a multiplication of only 7s.
11^s is a multiplication of only 11s.
2, 5, 7, 11 are all prime numbers. so, they have no other factors than 1 or themselves. especially not each other.
(a)
therefore, to find a we find how many times we can divide 1960 and the then resulting results by 2 :
1960/2 = 980
980/2 = 490
490/2 = 245
3 times, so 2³ is a factor, a = 3.
now, to find b we divide the remaining number of 245 by 5 as often as possible :
245/5 = 49
1 time, so 5¹ is a factor, b = 1.
to find c we divide the remaining number of 49 by 7 as often as possible :
49/7 = 7
7/7 = 1
2 times, and we are finished (there are no more factors to find in 1). so, 7² is a factor, c = 2.
(b)
1960 = 2³×5¹×7²
n = 2^p × 5^q × 7^r × 11^s
the LCM = 2³×5²×7²×11¹
the LCM (least common multiple) is the product of the longest chains of prime factors.
1960 provides already 2³, 5¹, 7² as chains.
what is missing compared to 2³×5²×7²×11¹ is 5¹(× the existing 5¹) and 11¹.
n does not provide any additional 2s or 7s as factors.
and so, n contributes to the LCM with the factors 5² and 11¹.
it could contain shorter chains of 2s and/ or 7s.
but to find the smallest n, we can ignore these.
the smallest possible n = 5²×11¹
which is 275.
