Re: For any integer k > 1, the term length of an integer refers to the n
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31 Jul 2023, 03:24
The term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to the integer. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3.
To find the maximum possible sum of the length of x and the length of y, given that x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, we need to find the values of x and y that maximize their lengths. One way to do this is to use the fact that the length of an integer is equal to the exponent of its prime factorization. For example, if k = 2^a * 3^b * 5^c * …, then the length of k is equal to a + b + c + …
Therefore, to maximize the length of an integer, we need to maximize its exponent. This means we need to choose the smallest possible base for each factor. Since we are dealing with positive integers, the smallest possible base is 2. So, we need to find x and y such that they are powers of 2 and satisfy the given inequality.
The largest power of 2 that is less than 1000 is 2^9 = 512. So, we can try x = 512 and see if we can find a suitable value for y. If x = 512, then x + 3y < 1000 implies that y < (1000 - 512) / 3 = 162.67. The largest power of 2 that is less than 162.67 is 2^7 = 128. So, we can try y = 128 and see if it satisfies the given inequality. If y = 128, then x + 3y = 512 + 3 * 128 = 896 < 1000. So, we have found a pair of values for x and y that satisfy the given inequality.
The length of x is equal to the exponent of its prime factorization, which is 9 in this case. The length of y is also equal to the exponent of its prime factorization, which is 7 in this case. So, the sum of the length of x and the length of y is equal to 9 + 7 = 16.
Therefore, the maximum possible sum of the length of x and the length of y is 16, when x = 512 and y = 128.