Find the smallest natural number n such that both sqrt of 85+n and sqrt of 140.n can be natural numbers

Answer 1

Prime factorization for 140 = 2²·5·7. Because 140·n became perfect square if n = 5·7 = 35. Then √140·n =

= 70 is natural bu 85 + 35 = 120 and √120 is not natural. Next n = 5·7·2² to make 140n = 140² perfect square.

Now 85 + n = 85 + 140 = 225 is perfect square also.

So, min n = 140

Answer 2

Hi Kayla,

I think the second condition is √(140 * n). We want to start with this number because we can find the exact values of n by prime factorizing 140, and we can find the prime factors on n to allow the square root of 140*n to be a natural number.

Prime factorization of 140: 140 = 2² * 5¹ * 7¹

For √(140 * n) to be a natural number, the exponents of the prime factors of 140 * n have to be even numbers. The minimum n that satisfies this condition is 5 * 7 = 35 becomes 140 * 35 = 2² * 5² * 7² (all the prime factors have even exponents), so

√(140 * n) = √(140 * 35) = √(2² * 5² * 7²) = 2 * 5 * 7 = 70, a natural number. The second condition is

√(85 + n) = √(85 + 35) = √120, which is not a natural number.

The next smallest n we can get is n = 2² * 5 * 7 = 140, because we have 140 * n = 140². Repeating the same thing as above,

√(140 * 140) = √(140²) = 140, a natural number

√(85 + n) = √(85 + 140) = √225 = 15, also a natural number.

So, the smallest natural number that fits both the conditions is n = 140.

Hope this helps! Please let me know if you have any questions.

Akshat Y.