A happy number is a positive integer defined in the following way: in

Question

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.

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

peachfuzz · Accepted Answer

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")

Answer: D

King407 · Answer

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?

ShristiK · Answer

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

So, the answer is A

Bunuel · Answer

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!

The correct answer is D.

bumpbot · Answer

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A happy number is a positive integer defined in the following way: in

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