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08 Jun 2023, 12:21
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Difficulty:
75%
(hard)
Question Stats:
59% (03:24) correct
41%
(02:43)
wrong
based on 85
sessions
History
Date
Time
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Not Attempted Yet
How many natural numbers upto 2050 are neither perfect squares nor perfect cubes?
(A) 1978
(B) 1996
(C) 1966
(D) 1990
(E) 1982
Posted from my mobile device
(A) 1978
(B) 1996
(C) 1966
(D) 1990
(E) 1982
Posted from my mobile device
08 Jun 2023, 12:46
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This problem doesn't much resemble GMAT items. "Natural number"—which just means "positive integer"—is not a term that you'll need to know for the GMAT, and the sheer amount of arithmetic work needed to solve this problem far exceeds anything GMAC would ever throw at you. Nonetheless, it can be useful as a practice item—especially if you're usually reluctant to use "low-tech" methods like longhand arithmetic.
Most of the integers from 1 to 2050 are neither perfect squares nor perfect cubes, so it should quickly become clear that the only reasonable way to solve this problem is to enumerate the values that ARE perfect squares and/or cubes, count them, and subtract that total from 2050.
Finding the numbers of perfect squares and perfect cubes that lie in the range 1-2050 is just trial and error (which, again, is the primary value-add of this problem: it encourages you to just dig in and do the arithmetic!) By just trying numbers until you find the threshold between ≤2050 and >2050, you'll eventually discover that...
• The first 45 perfect squares, from 1^2 = 1 to 45^2 = 2025, fall between 1 and 2050 (inclusive). Greater ones (from 46^2 = 2116 upward) do not.
• The first 12 perfect cubes, from 1^3 = 1 to 12^3 = 1728, fall between 1 and 2050 (inclusive). Greater ones (from 13^3 = 2197 upward) do not.
It would seem, then, that there are 57 values to exclude. However, the values that are BOTH perfect squares AND perfect cubes—in other words, perfect 6th powers—are counted twice.
These are just
• 1^6 = 1 (which is 1^2 and also 1^3);
• 2^6 = 64 (which is 8^2 and also 4^3); and
• 3^6 = 729 (which is 27^2 and also 9^3).
The next 6th power is 4^6, which would be equal to 64^2 or 16^3—both far beyond the desired range, as we can see by looking at the ranges of squares and cubes already found above.
So there are 3 duplicate values, meaning that there are 45 + 12 – 3 = 54 values to exclude. That leaves 2050 – 54 = 1996 values.
Most of the integers from 1 to 2050 are neither perfect squares nor perfect cubes, so it should quickly become clear that the only reasonable way to solve this problem is to enumerate the values that ARE perfect squares and/or cubes, count them, and subtract that total from 2050.
Finding the numbers of perfect squares and perfect cubes that lie in the range 1-2050 is just trial and error (which, again, is the primary value-add of this problem: it encourages you to just dig in and do the arithmetic!) By just trying numbers until you find the threshold between ≤2050 and >2050, you'll eventually discover that...
• The first 45 perfect squares, from 1^2 = 1 to 45^2 = 2025, fall between 1 and 2050 (inclusive). Greater ones (from 46^2 = 2116 upward) do not.
• The first 12 perfect cubes, from 1^3 = 1 to 12^3 = 1728, fall between 1 and 2050 (inclusive). Greater ones (from 13^3 = 2197 upward) do not.
It would seem, then, that there are 57 values to exclude. However, the values that are BOTH perfect squares AND perfect cubes—in other words, perfect 6th powers—are counted twice.
These are just
• 1^6 = 1 (which is 1^2 and also 1^3);
• 2^6 = 64 (which is 8^2 and also 4^3); and
• 3^6 = 729 (which is 27^2 and also 9^3).
The next 6th power is 4^6, which would be equal to 64^2 or 16^3—both far beyond the desired range, as we can see by looking at the ranges of squares and cubes already found above.
So there are 3 duplicate values, meaning that there are 45 + 12 – 3 = 54 values to exclude. That leaves 2050 – 54 = 1996 values.
08 Jun 2023, 12:50
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Arick
45 ^ 2 = 2025 and 46 ^ 2 = 2116 There are 45 perfect squares up to 2050.
12 ^ 3 = 1728 and 13 ^ 3 = 2197. There are 12 perfect cubes up to 2050 (Of these 134 ^ 3 and
9³ are squares).
Number of natural numbers up to 2050 which are either perfect squares or perfect cubes = 45 +12-3=54. Number of natural numbers up to 2050 which are neither perfect squares nor perfect cubes = 2050-54 = 1996.
