find the least number which when divided by 12,16,24,and 36 leaves a remainder 7 in each case.

Answer 1

Find the prime factorization of each integer.

12 = 2² ∙ 3

16 = 2⁴

24 = 2³ ∙ 3

36 = 2² ∙ 3²

The least common multiple (LCM) is the smallest number which can be exactly divided by each of the given number.

The LCM will be the product of multiplying the highest power of each prime number together.

In this case LCM = 2⁴ ∙ 3² = 16 ∙ 9 = 144

LCM for 12,16,24,36 = 144

Add a reminder 7

LCM + a reminder 7 = 144 + 7 = 151

The least number which when divided by 12,16,24,and 36 leaves a remainder 7 = 151

Proof:

151 / 12 = ( 144 + 7 ) / 12 = 144 / 12 + 7 / 12 = 12 + 7 / 12

A remainder = 7

151 / 16 = ( 144 + 7 ) / 16 = 144 / 16 + 7 / 16 = 9 + 7 / 16

A remainder = 7

151 / 24 = ( 144 + 7 ) / 24= 144 / 24 + 7 / 24 = 6 + 7 / 24

A remainder = 7

151 / 36 = ( 144 + 7 ) / 36= 144 / 36 + 7 / 36 = 4 + 7 / 36

A remainder = 7