Theory of Numbers Test - 3

1,573 = 11 × 143
= 11 × 11 × 13
So, highest factor = 11 × 13 = 143

Use the concept of power cycle, i.e. unit digit repeats after every power which is a multiple of 4.
99 has 24 complete cycles of 4 and 3 as remainder.
So, unit digit of (173)⁹⁹ will be the unit digit of 3³, which is 7.

270²⁷⁰ will have 270 zeroes at the end and 130¹³⁰ will have 130 zeroes at the end.

When these two are added, the rightmost non-zero numeral will be the rightmost non-zero numeral of 130¹³⁰, i.e. the last numeral of 3¹³⁰.

Since the last numeral of the power 3 repeats after every 4^th power, so the last numeral of 3¹³⁰ is equal to the last numeral of 3², i.e. 9.

The least number divisible by 3, 4, 5 and 7 = LCM of 3, 4, 5, 7 = 420.
Now, 420 is divisible by 3, 4, 5 and 7.
420 + 1 = 421 will always leave remainder 1, when divided by 3, 4, 5 or 7.
Required number = 420 + 1 = 421

L.C.M. of 6, 8, and 9 is 72.
∴ The required numbers = 72k + 5 (k is any natural number)
Put k = 6 and k = 7 to get the numbers between 400 and 550
∴ Required numbers = 72 × 6 + 5 =437 and 72 × 7 + 5 = 509

The numbers can be 14k₁ and 14k₂, where k₁ and k₂ do not have any common factor.
So, the possible two-digit numbers are 14 × 6 and 14 × 7.
∴ x + y = 84 + 98 = 182

Number of factors in 5005 can be found by using prime factorisation.
5005 = 5 × 7 × 11 × 13
Or 51 × 71 × 111 × 131
So, number of factors = 2 × 2 × 2 × 2 = 16
Hence, the total number of factors of 5005 = 16

1440 = 2⁵ × 3² × 5
To get the perfect square, we have to select one term from 2⁰, 2², 2⁴ and one term from 3⁰, 3²
and one term from 5⁰.
So, the answer is 3 × 2 × 1.
The answer is 6.

21 = 3 × 7
36 = 22 × 32
66 = 2 × 3 × 11
LCM of 21, 36, 66 = 2 × 2 × 3 × 3 × 7 × 11 = 2772
Multiples of 2772 less than 10,000 are 2772, 5544 and 8316.