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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # A happy number is a positive integer defined in the following way: in

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Math Expert V
Joined: 02 Sep 2009
Posts: 59075
A happy number is a positive integer defined in the following way: in  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 30% (02:41) correct 70% (02:49) wrong based on 146 sessions

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A happy number is a positive integer defined in the following way: in sequence A for integer i > 0, the term $$A_{(i+1)}$$ equals the sum of the squares of digits of $$A_i$$. If $$A_n = 1$$ for some positive integer n, then $$A_0$$ is happy. Which of the following is NOT happy?

A. 7
B. 10
C. 13
D. 16
E. 19

Kudos for a correct solution.

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Senior Manager  B
Joined: 28 Feb 2014
Posts: 289
Location: United States
Concentration: Strategy, General Management
Re: A happy number is a positive integer defined in the following way: in  [#permalink]

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Basically the prompt is just asking us which of the following answer choices does not equal 1 after squaring each of the respective digits and adding those results together.

All the other choices eventually result in 1. Only 16 does not do this.
16
1^2 + 6^2 = 37
3^2 + 7^2 = 58
5^2 + 8^2 = 89
8^2 + 9^2 = 145
1^2 + 4^2 + 5^2 = 42
4^2 + 2^2 = 20
2^2 + 0^2 = 4
4^2 = 16 (this sequence repeats and never results in 1, so 16 is never a "happy number")

Math Expert V
Joined: 02 Sep 2009
Posts: 59075
Re: A happy number is a positive integer defined in the following way: in  [#permalink]

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Bunuel wrote:
A happy number is a positive integer defined in the following way: in sequence A for integer i > 0, the term $$A_{(i+1)}$$ equals the sum of the squares of digits of $$A_i$$. If $$A_n = 1$$ for some positive integer n, then $$A_0$$ is happy. Which of the following is NOT happy?

A. 7
B. 10
C. 13
D. 16
E. 19

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

Since there can only be one right answer, only one of the choices can be unhappy; the other four numbers must be happy. So, as you check those numbers against the definition of happiness, make sure you see the same outcome for four of the numbers.

Start with answer choice (A), which is 7. If this is a happy number, then it should start a sequence that produces 1 eventually. We have the definition of the sequence; let’s apply it.

$$A_0 = 7$$
$$A_1$$ = sum of squares of digits of $$A_0 = 7^2 = 49$$
$$A_2$$ = sum of squares of digits of $$A_1 = 4^2 + 9^2 = 16 + 81 = 97$$
$$A_3$$ = sum of squares of digits of $$A_2 = 9^2 + 7^2 = 81 + 49 = 130$$
$$A_4$$ = sum of squares of digits of $$A_3 = 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10$$
$$A_5$$ = sum of squares of digits of $$A_4 = 1^2 + 0^2 = 1 + 0 = 1$$

You’ve gotten a 1, so now you know that 7 (the number you started with) is happy—and therefore not the right answer.

You can eliminate 10 (choice B) easily, either by directly applying the sequence definition or by observing that you got 10 along the way from 7 to 1, so 10 is happy as well.

Try 13 (choice C):

$$A_0 = 13$$
$$A_1$$ = sum of squares of digits of $$A_0 = 1^2 + 3^2 = 1 + 9 = 10$$
You can stop with this sequence now, since you know that 10 is happy. 13 is happy as well.

Try 16 (choice D):

$$A_0= 16$$
$$A_1$$ = sum of squares of digits of $$A_0 = 1^2 + 6^2 = 1 + 36 = 37$$
$$A_2$$ = sum of squares of digits of $$A_1 = 3^2 + 7^2 = 9 + 49 = 58$$
$$A_3$$ = sum of squares of digits of $$A_2 = 5^2 + 8^2 = 25 + 64 = 89$$
$$A_4$$ = sum of squares of digits of $$A_3 = 8^2 + 9^2 = 64 + 81 = 145$$
$$A_5$$ = sum of squares of digits of $$A_4 = 1^2 + 4^2 + 5^2 = 1 + 16 + 25 = 42$$
$$A_6$$ = sum of squares of digits of $$A_5 = 4^2 + 2^2 = 16 + 4 = 20$$
$$A_7$$ = sum of squares of digits of $$A_6 = 2^2 + 0^2 = 4$$
$$A_8$$ = sum of squares of digits of $$A_7 = 4^2 = 16$$

Uh-oh—you’re back to 16, so the cycle will repeat from here infinitely, never reaching 1. This means that 16 is unhappy, and thus is the answer.

If you check 19, you’ll find that it’s happy (19 → 82 → 68 → 100 → 1).

By the way, this definition of “happiness” really exists in number theory. We do make up some definitions for our problems, but this one’s “real,” so to speak!

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Re: A happy number is a positive integer defined in the following way: in  [#permalink]

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Bunuel wrote:
A happy number is a positive integer defined in the following way: in sequence A for integer i > 0, the term $$A_{(i+1)}$$ equals the sum of the squares of digits of A_i. If A_n = 1 for some positive integer n, then A_0 is happy. Which of the following is NOT happy?

A. 7
B. 10
C. 13
D. 16
E. 19

Kudos for a correct solution.

spent good 5 mins to figure out if there is a hidden trick or shortcut; no luck.

started plugging values.. the whole process of testing values again took more than 2 mins; picking digits and squaring and adding again and again.

Is there faster way to approach questions like this?
Intern  Joined: 26 Aug 2014
Posts: 42
GMAT 1: 650 Q49 V30 GMAT 2: 650 Q49 V31 WE: Programming (Computer Software)
Re: A happy number is a positive integer defined in the following way: in  [#permalink]

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Bunuel wrote:
A happy number is a positive integer defined in the following way: in sequence A for integer i > 0, the term $$A_{(i+1)}$$ equals the sum of the squares of digits of A_i. If A_n = 1 for some positive integer n, then A_0 is happy. Which of the following is NOT happy?

A. 7
B. 10
C. 13
D. 16
E. 19

Kudos for a correct solution.

Plugged in the choices to find out the answer.
Here is how I did it:

A) 7^2 = 49 => 4+9 = 13 => 1+3 =4 --> Not Happy
B) 1^2 +0^2 = 1+0 = 1 --> Happy number
C) 1^2 + 3^2 = 1+9 = 10 => 1+0 = 1 --> Happy number
D) 1^2 + 6^2 = 1+ 36 = 37 => 3+7 => 10 => 1+0 = 1 --> Happy number
E) 1^2 + 9^2 = 1+ 81 = 82 => 8+2 => 10 => 1+0 = 1 --> Happy number

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Re: A happy number is a positive integer defined in the following way: in  [#permalink]

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